mmtheorems3 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 201-300 - Page 3 of 107

Type	Label	Description
Statement
Theorem	sylbir 201	A mixed syllogism inference from a biconditional and an implication.
|- (ps <-> ph)   &   |- (ps -> ch)   =>   |- (ph -> ch)
Theorem	sylibd 202	A syllogism deduction.
|- (ph -> (ps -> ch))   &   |- (ph -> (ch <-> th))   =>   |- (ph -> (ps -> th))
Theorem	sylbid 203	A syllogism deduction.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch -> th))   =>   |- (ph -> (ps -> th))
Theorem	sylibrd 204	A syllogism deduction.
|- (ph -> (ps -> ch))   &   |- (ph -> (th <-> ch))   =>   |- (ph -> (ps -> th))
Theorem	sylbird 205	A syllogism deduction.
|- (ph -> (ch <-> ps))   &   |- (ph -> (ch -> th))   =>   |- (ph -> (ps -> th))
Theorem	syl5ib 206	A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition.
|- (ph -> (ps -> ch))   &   |- (th <-> ps)   =>   |- (ph -> (th -> ch))
Theorem	syl5ibr 207	A mixed syllogism inference from a nested implication and a biconditional.
|- (ph -> (ps -> ch))   &   |- (ps <-> th)   =>   |- (ph -> (th -> ch))
Theorem	syl5bi 208	A mixed syllogism inference.
|- (ph -> (ps <-> ch))   &   |- (th -> ps)   =>   |- (ph -> (th -> ch))
Theorem	syl5cbi 209	A mixed syllogism inference.
|- (ph -> (ps <-> ch))   &   |- (th -> ps)   =>   |- (th -> (ph -> ch))
Theorem	syl5bir 210	A mixed syllogism inference.
|- (ph -> (ps <-> ch))   &   |- (th -> ch)   =>   |- (ph -> (th -> ps))
Theorem	syl5cbir 211	A mixed syllogism inference.
|- (ph -> (ps <-> ch))   &   |- (th -> ch)   =>   |- (th -> (ph -> ps))
Theorem	syl6ib 212	A mixed syllogism inference from a nested implication and a biconditional.
|- (ph -> (ps -> ch))   &   |- (ch <-> th)   =>   |- (ph -> (ps -> th))
Theorem	syl6ibr 213	A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition.
|- (ph -> (ps -> ch))   &   |- (th <-> ch)   =>   |- (ph -> (ps -> th))
Theorem	syl6bi 214	A mixed syllogism inference.
|- (ph -> (ps <-> ch))   &   |- (ch -> th)   =>   |- (ph -> (ps -> th))
Theorem	syl6bir 215	A mixed syllogism inference.
|- (ph -> (ch <-> ps))   &   |- (ch -> th)   =>   |- (ph -> (ps -> th))
Theorem	syl7ib 216	A mixed syllogism inference from a doubly nested implication and a biconditional.
|- (ph -> (ps -> (ch -> th)))   &   |- (ta <-> ch)   =>   |- (ph -> (ps -> (ta -> th)))
Theorem	syl8ib 217	A syllogism rule of inference. The second premise is used to replace the consequent of the first premise.
|- (ph -> (ps -> (ch -> th)))   &   |- (th <-> ta)   =>   |- (ph -> (ps -> (ch -> ta)))
Theorem	3imtr3 218	A mixed syllogism inference, useful for removing a definition from both sides of an implication.
|- (ph -> ps)   &   |- (ph <-> ch)   &   |- (ps <-> th)   =>   |- (ch -> th)
Theorem	3imtr4 219	A mixed syllogism inference, useful for applying a definition to both sides of an implication.
|- (ph -> ps)   &   |- (ch <-> ph)   &   |- (th <-> ps)   =>   |- (ch -> th)
Theorem	con1bii 220	A contraposition inference.
|- (-. ph <-> ps)   =>   |- (-. ps <-> ph)
Theorem	con2bii 221	A contraposition inference.
|- (ph <-> -. ps)   =>   |- (ps <-> -. ph)
Logical disjunction and conjunction
Syntax	wo 222	Extend wff definition to include disjunction ('or').
wff (ph \/ ps)
Syntax	wa 223	Extend wff definition to include conjunction ('and').
wff (ph /\ ps)
Definition	df-or 224	Define disjunction (logical 'or'). This is our first use of the biconditional connective in a definition; we use it in place of the traditional "<=def=>", which means the same thing, except that we can manipulate the biconditional connective directly in proofs rather than having to rely on an informal definition substitution rule. Note that if we mechanically substitute (-. ph -> ps) for (ph \/ ps), we end up with an instance of previously proved theorem pm4.2 170. This is the justification for the definition, along with the fact that it introduces a new symbol \/. Definition of [Margaris] p. 49.
|- ((ph \/ ps) <-> (-. ph -> ps))
Definition	df-an 225	Define conjunction (logical 'and'). Definition of [Margaris] p. 49.
|- ((ph /\ ps) <-> -. (ph -> -. ps))
Theorem	pm4.64 226	Theorem *4.64 of [WhiteheadRussell] p. 120.
|- ((-. ph -> ps) <-> (ph \/ ps))
Theorem	pm2.54 227	Theorem *2.54 of [WhiteheadRussell] p. 107.
|- ((-. ph -> ps) -> (ph \/ ps))
Theorem	pm4.63 228	Theorem *4.63 of [WhiteheadRussell] p. 120.
|- (-. (ph -> -. ps) <-> (ph /\ ps))
Theorem	dfor2 229	Logical 'or' expressed in terms of implication only. Theorem *5.25 of [WhiteheadRussell] p. 124.
|- ((ph \/ ps) <-> ((ph -> ps) -> ps))
Theorem	ori 230	Infer implication from disjunction.
|- (ph \/ ps)   =>   |- (-. ph -> ps)
Theorem	orri 231	Infer implication from disjunction.
|- (-. ph -> ps)   =>   |- (ph \/ ps)
Theorem	ord 232	Deduce implication from disjunction.
|- (ph -> (ps \/ ch))   =>   |- (ph -> (-. ps -> ch))
Theorem	orrd 233	Deduce implication from disjunction.
|- (ph -> (-. ps -> ch))   =>   |- (ph -> (ps \/ ch))
Theorem	imor 234	Implication in terms of disjunction. Theorem *4.6 of [WhiteheadRussell] p. 120.
|- ((ph -> ps) <-> (-. ph \/ ps))
Theorem	pm4.62 235	Theorem *4.62 of [WhiteheadRussell] p. 120.
|- ((ph -> -. ps) <-> (-. ph \/ -. ps))
Theorem	pm4.66 236	Theorem *4.66 of [WhiteheadRussell] p. 120.
|- ((-. ph -> -. ps) <-> (ph \/ -. ps))
Theorem	iman 237	Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176.
|- ((ph -> ps) <-> -. (ph /\ -. ps))
Theorem	annim 238	Express conjunction in terms of implication.
|- ((ph /\ -. ps) <-> -. (ph -> ps))
Theorem	pm4.61 239	Theorem *4.61 of [WhiteheadRussell] p. 120.
|- (-. (ph -> ps) <-> (ph /\ -. ps))
Theorem	pm4.65 240	Theorem *4.65 of [WhiteheadRussell] p. 120.
|- (-. (-. ph -> ps) <-> (-. ph /\ -. ps))
Theorem	pm4.67 241	Theorem *4.67 of [WhiteheadRussell] p. 120.
|- (-. (-. ph -> -. ps) <-> (-. ph /\ ps))
Theorem	imnan 242	Express implication in terms of conjunction.
|- ((ph -> -. ps) <-> -. (ph /\ ps))
Theorem	oridm 243	Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell] p. 117.
|- ((ph \/ ph) <-> ph)
Theorem	pm4.25 244	Theorem *4.25 of [WhiteheadRussell] p. 117.
|- (ph <-> (ph \/ ph))
Theorem	pm1.2 245	Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96.
|- ((ph \/ ph) -> ph)
Theorem	orcom 246	Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell] p. 118.
|- ((ph \/ ps) <-> (ps \/ ph))
Theorem	pm1.4 247	Axiom *1.4 of [WhiteheadRussell] p. 96.
|- ((ph \/ ps) -> (ps \/ ph))
Theorem	pm2.62 248	Theorem *2.62 of [WhiteheadRussell] p. 107.
|- ((ph \/ ps) -> ((ph -> ps) -> ps))
Theorem	pm2.621 249	Theorem *2.621 of [WhiteheadRussell] p. 107.
|- ((ph -> ps) -> ((ph \/ ps) -> ps))
Theorem	pm2.68 250	Theorem *2.68 of [WhiteheadRussell] p. 108.
|- (((ph -> ps) -> ps) -> (ph \/ ps))
Theorem	orel1 251	Elimination of disjunction by denial of a disjunct. Theorem *2.55 of [WhiteheadRussell] p. 107.
|- (-. ph -> ((ph \/ ps) -> ps))
Theorem	orel2 252	Elimination of disjunction by denial of a disjunct. Theorem *2.56 of [WhiteheadRussell] p. 107.
|- (-. ph -> ((ps \/ ph) -> ps))
Theorem	pm2.25 253	Theorem *2.25 of [WhiteheadRussell] p. 104.
|- (ph \/ ((ph \/ ps) -> ps))
Theorem	pm2.53 254	Theorem *2.53 of [WhiteheadRussell] p. 107.
|- ((ph \/ ps) -> (-. ph -> ps))
Theorem	orbi2i 255	Inference adding a left disjunct to both sides of a logical equivalence.
|- (ph <-> ps)   =>   |- ((ch \/ ph) <-> (ch \/ ps))
Theorem	orbi1i 256	Inference adding a right disjunct to both sides of a logical equivalence.
|- (ph <-> ps)   =>   |- ((ph \/ ch) <-> (ps \/ ch))
Theorem	orbi12i 257	Infer the disjunction of two equivalences.
|- (ph <-> ps)   &   |- (ch <-> th)   =>   |- ((ph \/ ch) <-> (ps \/ th))
Theorem	or12 258	A rearrangement of disjuncts.
|- ((ph \/ (ps \/ ch)) <-> (ps \/ (ph \/ ch)))
Theorem	pm1.5 259	Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96.
|- ((ph \/ (ps \/ ch)) -> (ps \/ (ph \/ ch)))
Theorem	orass 260	Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell] p. 118.
|- (((ph \/ ps) \/ ch) <-> (ph \/ (ps \/ ch)))
Theorem	pm2.31 261	Theorem *2.31 of [WhiteheadRussell] p. 104.
|- ((ph \/ (ps \/ ch)) -> ((ph \/ ps) \/ ch))
Theorem	pm2.32 262	Theorem *2.32 of [WhiteheadRussell] p. 105.
|- (((ph \/ ps) \/ ch) -> (ph \/ (ps \/ ch)))
Theorem	or23 263	A rearrangement of disjuncts.
|- (((ph \/ ps) \/ ch) <-> ((ph \/ ch) \/ ps))
Theorem	or4 264	Rearrangement of 4 disjuncts.
|- (((ph \/ ps) \/ (ch \/ th)) <-> ((ph \/ ch) \/ (ps \/ th)))
Theorem	or42 265	Rearrangement of 4 disjuncts.
|- (((ph \/ ps) \/ (ch \/ th)) <-> ((ph \/ ch) \/ (th \/ ps)))
Theorem	orordi 266	Distribution of disjunction over disjunction.
|- ((ph \/ (ps \/ ch)) <-> ((ph \/ ps) \/ (ph \/ ch)))
Theorem	orordir 267	Distribution of disjunction over disjunction.
|- (((ph \/ ps) \/ ch) <-> ((ph \/ ch) \/ (ps \/ ch)))
Theorem	olc 268	Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96.
|- (ph -> (ps \/ ph))
Theorem	orc 269	Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104.
|- (ph -> (ph \/ ps))
Theorem	orci 270	Deduction introducing a disjunct.
|- ph   =>   |- (ph \/ ps)
Theorem	olci 271	Deduction introducing a disjunct.