mmtheorems10 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 901-1000 - Page 10 of 107

Type	Label	Description
Statement
Theorem	mp3an1 901	An inference based on modus ponens.
|- ph   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ps /\ ch) -> th)
Theorem	mp3an2 902	An inference based on modus ponens.
|- ps   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mp3an3 903	An inference based on modus ponens.
|- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mp3an12 904	An inference based on modus ponens.
|- ph   &   |- ps   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ch -> th)
Theorem	mp3an13 905	An inference based on modus ponens.
|- ph   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ps -> th)
Theorem	mp3an23 906	An inference based on modus ponens.
|- ps   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> th)
Theorem	mp3an1i 907	An inference based on modus ponens.
|- ps   &   |- (ph -> ((ps /\ ch /\ th) -> ta))   =>   |- (ph -> ((ch /\ th) -> ta))
Theorem	mp3anl1 908	An inference based on modus ponens.
|- ph   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ps /\ ch) /\ th) -> ta)
Theorem	mp3anl2 909	An inference based on modus ponens.
|- ps   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ch) /\ th) -> ta)
Theorem	mp3anl3 910	An inference based on modus ponens.
|- ch   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ps) /\ th) -> ta)
Theorem	mp3anr1 911	An inference based on modus ponens.
|- ps   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ch /\ th)) -> ta)
Theorem	mp3anr2 912	An inference based on modus ponens.
|- ch   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ th)) -> ta)
Theorem	mp3anr3 913	An inference based on modus ponens.
|- th   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ ch)) -> ta)
Theorem	mp3an 914	An inference based on modus ponens.
|- ph   &   |- ps   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- th
Theorem	mpd3an3 915	An inference based on modus ponens.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mpd3an23 916	An inference based on modus ponens.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> th)
Theorem	biimp3a 917	Infer implication from a logical equivalence. Similar to biimpa 416.
|- ((ph /\ ps) -> (ch <-> th))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3anandis 918	Inference that undistributes a triple conjunction in the antecedent.
|- (((ph /\ ps) /\ (ph /\ ch) /\ (ph /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ ch /\ th)) -> ta)
Theorem	3anandirs 919	Inference that undistributes a triple conjunction in the antecedent.
|- (((ph /\ th) /\ (ps /\ th) /\ (ch /\ th)) -> ta)   =>   |- (((ph /\ ps /\ ch) /\ th) -> ta)
Theorem	ecase23d 920	Deduction for elimination by cases.
|- (ph -> -. ch)   &   |- (ph -> -. th)   &   |- (ph -> (ps \/ ch \/ th))   =>   |- (ph -> ps)
Theorem	3ecase 921	Inference for elimination by cases.
|- (-. ph -> th)   &   |- (-. ps -> th)   &   |- (-. ch -> th)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- th
Other axiomatizations of classical propositional calculus
Theorem	meredith 922	Carew Meredith's sole axiom for propositional calculus. This amazing formula is thought to be the shortest possible single axiom for propositional calculus with inference rule ax-mp 7, where negation and implication are primitive. Here we prove Meredith's axiom from ax-1 4, ax-2 5, and ax-3 6. Then from it we derive the Lukasiewicz axioms luk-1 936, luk-2 937, and luk-3 938. Using these we finally re-derive our axioms as ax1 947, ax2 948, and ax3 949, thus proving the equivalence of all three systems. C. A. Meredith, "Single Axioms for the Systems (C,N), (C,O) and (A,N) of the Two-Valued Propositional Calculus," The Journal of Computing Systems vol. 3 (1953), pp. 155-164. Meredith claimed to be close to a proof that this axiom is the shortest possible, but the proof was apparently never completed. An obscure Irish lecturer, Meredith (1904-1976) became enamored with logic somewhat late in life after attending talks by Lukasiewicz and produced many remarkable results such as this axiom. From his obituary: "He did logic whenever time and opportunity presented themselves, and he did it on whatever materials came to hand: in a pub, his favored pint of porter within reach, he would use the inside of cigarette packs to write proofs for logical colleagues."
|- (((((ph -> ps) -> (-. ch -> -. th)) -> ch) -> ta) -> ((ta -> ph) -> (th -> ph)))
Theorem	merlem1 923	Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.)
|- (((ch -> (-. ph -> ps)) -> ta) -> (ph -> ta))
Theorem	merlem2 924	Step 4 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ph -> ph) -> ch) -> (th -> ch))
Theorem	merlem3 925	Step 7 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ps -> ch) -> ph) -> (ch -> ph))
Theorem	merlem4 926	Step 8 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (ta -> ((ta -> ph) -> (th -> ph)))
Theorem	merlem5 927	Step 11 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> ps) -> (-. -. ph -> ps))
Theorem	merlem6 928	Step 12 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (ch -> (((ps -> ch) -> ph) -> (th -> ph)))
Theorem	merlem7 929	Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (ph -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)))
Theorem	merlem8 930	Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th))
Theorem	merlem9 931	Step 18 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ph -> ps) -> (ch -> (th -> (ps -> ta)))) -> (et -> (ch -> (th -> (ps -> ta)))))
Theorem	merlem10 932	Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> (ph -> ps)) -> (th -> (ph -> ps)))
Theorem	merlem11 933	Step 20 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> (ph -> ps)) -> (ph -> ps))
Theorem	merlem12 934	Step 28 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((th -> (-. -. ch -> ch)) -> ph) -> ph)
Theorem	merlem13 935	Step 35 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> ps) -> (((th -> (-. -. ch -> ch)) -> -. -. ph) -> ps))
Theorem	luk-1 936	1 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.
|- ((ph -> ps) -> ((ps -> ch) -> (ph -> ch)))
Theorem	luk-2 937	2 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.
|- ((-. ph -> ph) -> ph)
Theorem	luk-3 938	3 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.
|- (ph -> (-. ph -> ps))
Theorem	luklem1 939	Used to rederive standard propositional axioms from Lukasiewicz'.
|- (ph -> ps)   &   |- (ps -> ch)   =>   |- (ph -> ch)
Theorem	luklem2 940	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> -. ps) -> (((ph -> ch) -> th) -> (ps -> th)))
Theorem	luklem3 941	Used to rederive standard propositional axioms from Lukasiewicz'.
|- (ph -> (((-. ph -> ps) -> ch) -> (th -> ch)))
Theorem	luklem4 942	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((((-. ph -> ph) -> ph) -> ps) -> ps)
Theorem	luklem5 943	Used to rederive standard propositional axioms from Lukasiewicz'.
|- (ph -> (ps -> ph))
Theorem	luklem6 944	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> (ph -> ps)) -> (ph -> ps))
Theorem	luklem7 945	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> (ps -> ch)) -> (ps -> (ph -> ch)))
Theorem	luklem8 946	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> ps) -> ((ch -> ph) -> (ch -> ps)))
Theorem	ax1 947	Standard propositional axiom derived from Lukasiewicz axioms.
|- (ph -> (ps -> ph))
Theorem	ax2 948	Standard propositional axiom derived from Lukasiewicz axioms.
|- ((ph -> (ps -> ch)) -> ((ph -> ps) -> (ph -> ch)))
Theorem	ax3 949	Standard propositional axiom derived from Lukasiewicz axioms.
|- ((-. ph -> -. ps) -> (ps -> ph))
Theorem	nicodraw 950	Axiom of Nicod from Introduction to Mathematical Philosophy B. Russell, p. 152. The axiom is recovered from this raw form by substituting (ph | ps) for -. (ph /\ ps), where | is the Sheffer stroke (NAND) connective, so that the Sheffer stroke becomes the sole connective. See nicodmpraw 951 for the inference rule. (Based on a proof by Jeff Hoffman, 19-Nov-2007.)
|- -. (-. (ph /\ -. (ch /\ ps)) /\ -. (-. (ta /\ -. (ta /\ ta)) /\ -. (-. (th /\ ch) /\ -. (-. (ph /\ th) /\ -. (ph /\ th)))))
Theorem	nicodmpraw 951	The inference rule for the axiom of Nicod, in raw form as explained in nicodraw 950.
|- ph   &   |- -. (ph /\ -. (ch /\ ps))   =>   |- ps
Predicate calculus axiomatization
The axioms of predicate calculus
Syntax	wal 952	Extend wff definition to include the universal quantifier ('for all'). A.xph is read "ph (phi) is true for all x." Typically, in its final application ph would be replaced with a wff containing a (free) occurrence of the variable x, for example x = y. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of x. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same.
wff A.xph
Syntax	cv 953	This syntax construction states that a variable x, which has been declared to be a set variable by $f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder {y | y e. x} is a class by cab 1461. Since (when y is distinct from x) we have x = {y | y e. x} by cvjust 1469, we can argue that that the syntax "class x" can be viewed as an abbreviation for "class {y | y e. x}". See the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." While it is tempting and perhaps occasionally useful to view cv 953 as a "type conversion" from a set variable to a class variable, keep in mind that cv 953 is intrinsically no different from any other class-building syntax such as cab 1461, cun 2041, or c0 2276. (The purpose of introducing class x here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 955 of predicate calculus from the wceq 954 of set theory, so that we don't "overload" the = connective with two syntax definitions. This is done to prevent ambiguity that causes problems in some Metamath parsers. The remaining part of this description applies to set theory, not predicate calculus.)
class x
Syntax	wceq 954	Extend wff definition to include class equality. (The purpose of introducing wff A = B here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 955 of predicate calculus in terms of the wceq 954 of set theory, so that we don't "overload" the = connective with two syntax definitions. This is done to prevent ambiguity that causes problems in some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the = in x = y could be the = of either weq 955 or wceq 954, although mathematically it makes no difference. The class variables A and B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 1467 for more information on the set theory usage of wceq 954.)
wff A = B
Theorem	weq 955	Extend wff definition to include atomic formulas using the equality predicate. (Instead of introducing weq 955 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wceq 954. This lets us avoid overloading the = connective, thus preventing ambiguity that causes problems in certain Metamath parsers. However, logically weq 955 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 954. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.)
wff x = y
Syntax	wcel 956	Extend wff definition to include the membership conn