mmtheorems14 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 1301-1400 - Page 14 of 107

Type	Label	Description
Statement
Theorem	19.37v 1301	Special case of Theorem 19.37 of [Margaris] p. 90.
|- (E.x(ph -> ps) <-> (ph -> E.xps))
Theorem	19.37aiv 1302	Inference from Theorem 19.37 of [Margaris] p. 90.
|- E.x(ph -> ps)   =>   |- (ph -> E.xps)
Theorem	19.41v 1303	Special case of Theorem 19.41 of [Margaris] p. 90.
|- (E.x(ph /\ ps) <-> (E.xph /\ ps))
Theorem	19.41vv 1304	Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers.
|- (E.xE.y(ph /\ ps) <-> (E.xE.yph /\ ps))
Theorem	19.41vvv 1305	Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers.
|- (E.xE.yE.z(ph /\ ps) <-> (E.xE.yE.zph /\ ps))
Theorem	19.42v 1306	Special case of Theorem 19.42 of [Margaris] p. 90.
|- (E.x(ph /\ ps) <-> (ph /\ E.xps))
Theorem	exdistr 1307	Distribution of existential quantifiers.
|- (E.xE.y(ph /\ ps) <-> E.x(ph /\ E.yps))
Theorem	19.42vv 1308	Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers.
|- (E.xE.y(ph /\ ps) <-> (ph /\ E.xE.yps))
Theorem	exdistr2 1309	Distribution of existential quantifiers.
|- (E.xE.yE.z(ph /\ ps) <-> E.x(ph /\ E.yE.zps))
Theorem	3exdistr 1310	Distribution of existential quantifiers.
|- (E.xE.yE.z(ph /\ ps /\ ch) <-> E.x(ph /\ E.y(ps /\ E.zch)))
Theorem	4exdistr 1311	Distribution of existential quantifiers.
|- (E.xE.yE.zE.w((ph /\ ps) /\ (ch /\ th)) <-> E.x(ph /\ E.y(ps /\ E.z(ch /\ E.wth))))
Theorem	cbvalv 1312	Rule used to change bound variables with implicit substitution.
|- (x = y -> (ph <-> ps))   =>   |- (A.xph <-> A.yps)
Theorem	cbvexv 1313	Rule used to change bound variables with implicit substitution.
|- (x = y -> (ph <-> ps))   =>   |- (E.xph <-> E.yps)
Theorem	cbval2 1314	Rule used to change bound variables with implicit substitution.
|- (ph -> A.zph)   &   |- (ph -> A.wph)   &   |- (ps -> A.xps)   &   |- (ps -> A.yps)   &   |- ((x = z /\ y = w) -> (ph <-> ps))   =>   |- (A.xA.yph <-> A.zA.wps)
Theorem	cbvex2 1315	Rule used to change bound variables with implicit substitution.
|- (ph -> A.zph)   &   |- (ph -> A.wph)   &   |- (ps -> A.xps)   &   |- (ps -> A.yps)   &   |- ((x = z /\ y = w) -> (ph <-> ps))   =>   |- (E.xE.yph <-> E.zE.wps)
Theorem	cbval2v 1316	Rule used to change bound variables with implicit substitution.
|- ((x = z /\ y = w) -> (ph <-> ps))   =>   |- (A.xA.yph <-> A.zA.wps)
Theorem	cbvex2v 1317	Rule used to change bound variables with implicit substitution.
|- ((x = z /\ y = w) -> (ph <-> ps))   =>   |- (E.xE.yph <-> E.zE.wps)
Theorem	cbvald 1318	Deduction used to change bound variables with implicit substitution, particularly useful in conjunction with dvelim 1350.
|- (ph -> A.yph)   &   |- (ph -> (ps -> A.yps))   &   |- (ph -> (x = y -> (ps <-> ch)))   =>   |- (ph -> (A.xps <-> A.ych))
Theorem	cbvexd 1319	Deduction used to change bound variables with implicit substitution, particularly useful in conjunction with dvelim 1350.
|- (ph -> A.yph)   &   |- (ph -> (ps -> A.yps))   &   |- (ph -> (x = y -> (ps <-> ch)))   =>   |- (ph -> (E.xps <-> E.ych))
Theorem	cbvex4v 1320	Rule used to change bound variables with implicit substitution.
|- ((x = v /\ y = u) -> (ph <-> ps))   &   |- ((z = f /\ w = g) -> (ps <-> ch))   =>   |- (E.xE.yE.zE.wph <-> E.vE.uE.fE.gch)
Theorem	eeanv 1321	Rearrange existential quantifiers.
|- (E.xE.y(ph /\ ps) <-> (E.xph /\ E.yps))
Theorem	eeeanv 1322	Rearrange existential quantifiers.
|- (E.xE.yE.z(ph /\ ps /\ ch) <-> (E.xph /\ E.yps /\ E.zch))
Theorem	ee4anv 1323	Rearrange existential quantifiers.
|- (E.xE.yE.zE.w(ph /\ ps) <-> (E.xE.yph /\ E.zE.wps))
Theorem	nexdv 1324	Deduction for generalization rule for negated wff.
|- (ph -> -. ps)   =>   |- (ph -> -. E.xps)
Theorem	chvarv 1325	Implicit substitution of y for x into a theorem.
|- (x = y -> (ph <-> ps))   &   |- ph   =>   |- ps
Theorem	cleljust 1326	When the class variables in definition df-clel 1470 are replaced with set variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the set variables in wel 957 with the class variables in wcel 956.
|- (x e. y <-> E.z(z = x /\ z e. y))
More substitution theorems
Theorem	equsb3lem 1327	Lemma for equsb3 1328.
Theorem	equsb3 1328	Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
|- ([x / y]y = z <-> x = z)
Theorem	elsb3 1329	Substitution applied to an atomic membership wff.
|- ([x / y]y e. z <-> x e. z)
Theorem	hbs1 1330	x is not free in [y / x]ph when x and y are distinct.
|- ([y / x]ph -> A.x[y / x]ph)
Theorem	hbsb 1331	If z is not free in ph, it is not free in [y / x]ph when y and z are distinct.
|- (ph -> A.zph)   =>   |- ([y / x]ph -> A.z[y / x]ph)
Theorem	sbcom2 1332	Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint).
|- ([w / z][y / x]ph <-> [y / x][w / z]ph)
Theorem	2sb5 1333	Equivalence for double substitution.
|- ([z / x][w / y]ph <-> E.xE.y((x = z /\ y = w) /\ ph))
Theorem	2sb6 1334	Equivalence for double substitution.
|- ([z / x][w / y]ph <-> A.xA.y((x = z /\ y = w) -> ph))
Theorem	sb6a 1335	Equivalence for substitution.
|- ([y / x]ph <-> A.x(x = y -> [x / y]ph))
Theorem	2sb5rf 1336	Reversed double substitution.
|- (ph -> A.zph)   &   |- (ph -> A.wph)   =>   |- (ph <-> E.zE.w((z = x /\ w = y) /\ [z / x][w / y]ph))
Theorem	2sb6rf 1337	Reversed double substitution.
|- (ph -> A.zph)   &   |- (ph -> A.wph)   =>   |- (ph <-> A.zA.w((z = x /\ w = y) -> [z / x][w / y]ph))
Theorem	dfsb7 1338	An alternate definition of proper substitution df-sb 1170. By introducing a dummy variable z in the definiens, we are able to eliminate any distinct variable restrictions among the variables x, y, and ph of the definiendum. No distinct variable conflicts arise because z effectively insulates x from y. To achieve this, we use a chain of two substitutions in the form of sb5 1266, first z for x then y for z. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 1462. Theorem sb7f 1339 provides a version where ph and z don't have to be distinct.
|- ([y / x]ph <-> E.z(z = y /\ E.x(x = z /\ ph)))
Theorem	sb7f 1339	This version of dfsb7 1338 does not require that ph and z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 969 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1170 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.)
|- (ph -> A.zph)   =>   |- ([y / x]ph <-> E.z(z = y /\ E.x(x = z /\ ph)))
Theorem	sb10f 1340	Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived.
|- (ph -> A.xph)   =>   |- ([y / z]ph <-> E.x(x = y /\ [x / z]ph))
Theorem	sbid2v 1341	An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint).
|- ([y / x][x / y]ph <-> ph)
Theorem	sbelx 1342	Elimination of substitution.
|- (ph <-> E.x(x = y /\ [x / y]ph))
Theorem	sbel2x 1343	Elimination of double substitution.
|- (ph <-> E.xE.y((x = z /\ y = w) /\ [y / w][x / z]ph))
Theorem	sbal1 1344	A theorem used in elimination of disjoint variable restriction on x and y by replacing it with a distinctor -. A.xx = z.
|- (-. A.x x = z -> ([z / y]A.xph <-> A.x[z / y]ph))
Theorem	sbal 1345	Move universal quantifier in and out of substitution.
|- ([z / y]A.xph <-> A.x[z / y]ph)
Theorem	sbex 1346	Move existential quantifier in and out of substitution.
|- ([z / y]E.xph <-> E.x[z / y]ph)
Theorem	sbalv 1347	Quantify with new variable inside substitution.
|- ([y / x]ph <-> ps)   =>   |- ([y / x]A.zph <-> A.zps)
Theorem	exsb 1348	An equivalent expression for existence.
|- (E.xph <-> E.yA.x(x = y -> ph))
Theorem	2exsb 1349	An equivalent expression for double existence.
|- (E.xE.yph <-> E.zE.wA.xA.y((x = z /\ y = w) -> ph))
Theorem	dvelim 1350	This theorem can be used to eliminate a distinct variable restriction on x and z and replace it with the "distinctor" -. A.xx = y as an antecedent. ph normally has z free and can be read ph(z), and ps substitutes y for z and can be read ph(y). We don't require that x and y be distinct: if they aren't, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent. To obtain a closed-theorem form of this inference, prefix the hypotheses with A.xA.z, conjoin them, and apply dvelimdf 1249.
|- (ph -> A.xph)   &   |- (z = y -> (ph <-> ps))   =>   |- (-. A.x x = y -> (ps -> A.xps))
Theorem	dvelimALT 1351	Version of dvelim 1350 that doesn't use ax-10 964. (See dvelimfALT 1151 for a version that doesn't use ax-11 965.)
|- (ph -> A.xph)   &   |- (z = y -> (ph <-> ps))   =>