P. 17 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 1601-1700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	19.12 1601	Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.)
|- ( E. x A. y ph -> A. y E. x ph )
Theorem	19.19 1602	Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- F/ x ph   =>    |- ( A. x ( ph <-> ps ) -> ( ph <-> E. x ps ) )
Theorem	19.21-2 1603	Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)
|- F/ x ph   &    |- F/ y ph   =>    |- ( A. x A. y ( ph -> ps ) <-> ( ph -> A. x A. y ps ) )
Theorem	nf2 1604	An alternate definition of df-nf 1396, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- ( F/ x ph <-> ( E. x ph -> A. x ph ) )
Theorem	nf3 1605	An alternate definition of df-nf 1396. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- ( F/ x ph <-> A. x ( E. x ph -> ph ) )
Theorem	nf4dc 1606	Variable x is effectively not free in ph iff ph is always true or always false, given a decidability condition. The reverse direction, nf4r 1607, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
|- (DECID E. x ph -> ( F/ x ph <-> ( A. x ph \/ A. x -. ph ) ) )
Theorem	nf4r 1607	If ph is always true or always false, then variable x is effectively not free in ph. The converse holds given a decidability condition, as seen at nf4dc 1606. (Contributed by Jim Kingdon, 21-Jul-2018.)
|- ( ( A. x ph \/ A. x -. ph ) -> F/ x ph )
Theorem	19.36i 1608	Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
|- F/ x ps   &    |- E. x ( ph -> ps )   =>    |- ( A. x ph -> ps )
Theorem	19.36-1 1609	Closed form of 19.36i 1608. One direction of Theorem 19.36 of [Margaris] p. 90. The converse holds in classical logic, but does not hold (for all propositions) in intuitionistic logic. (Contributed by Jim Kingdon, 20-Jun-2018.)
|- F/ x ps   =>    |- ( E. x ( ph -> ps ) -> ( A. x ph -> ps ) )
Theorem	19.37-1 1610	One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.)
|- F/ x ph   =>    |- ( E. x ( ph -> ps ) -> ( ph -> E. x ps ) )
Theorem	19.37aiv 1611*	Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- E. x ( ph -> ps )   =>    |- ( ph -> E. x ps )
Theorem	19.38 1612	Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
Theorem	19.23t 1613	Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
|- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )
Theorem	19.23 1614	Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
|- F/ x ps   =>    |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) )
Theorem	19.32dc 1615	Theorem 19.32 of [Margaris] p. 90, where ph is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
|- F/ x ph   =>    |- (DECID ph -> ( A. x ( ph \/ ps ) <-> ( ph \/ A. x ps ) ) )
Theorem	19.32r 1616	One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if ph is decidable, as seen at 19.32dc 1615. (Contributed by Jim Kingdon, 28-Jul-2018.)
|- F/ x ph   =>    |- ( ( ph \/ A. x ps ) -> A. x ( ph \/ ps ) )
Theorem	19.31r 1617	One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
|- F/ x ps   =>    |- ( ( A. x ph \/ ps ) -> A. x ( ph \/ ps ) )
Theorem	19.44 1618	Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- F/ x ps   =>    |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ ps ) )
Theorem	19.45 1619	Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- F/ x ph   =>    |- ( E. x ( ph \/ ps ) <-> ( ph \/ E. x ps ) )
Theorem	19.34 1620	Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ( A. x ph \/ E. x ps ) -> E. x ( ph \/ ps ) )
Theorem	19.41h 1621	Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1622 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
|- ( ps -> A. x ps )   =>    |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) )
Theorem	19.41 1622	Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
|- F/ x ps   =>    |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) )
Theorem	19.42h 1623	Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1624 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)
|- ( ph -> A. x ph )   =>    |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )
Theorem	19.42 1624	Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
|- F/ x ph   =>    |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )
Theorem	excom13 1625	Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
|- ( E. x E. y E. z ph <-> E. z E. y E. x ph )
Theorem	exrot3 1626	Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
|- ( E. x E. y E. z ph <-> E. y E. z E. x ph )
Theorem	exrot4 1627	Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
|- ( E. x E. y E. z E. w ph <-> E. z E. w E. x E. y ph )
Theorem	nexr 1628	Inference from 19.8a 1528. (Contributed by Jeff Hankins, 26-Jul-2009.)
|- -. E. x ph   =>    |- -. ph
Theorem	exan 1629	Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( E. x ph /\ ps )   =>    |- E. x ( ph /\ ps )
Theorem	hbexd 1630	Deduction form of bound-variable hypothesis builder hbex 1573. (Contributed by NM, 2-Jan-2002.)
|- ( ph -> A. y ph )   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> ( E. y ps -> A. x E. y ps ) )
Theorem	eeor 1631	Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
|- F/ y ph   &    |- F/ x ps   =>    |- ( E. x E. y ( ph \/ ps ) <-> ( E. x ph \/ E. y ps ) )
1.3.8  Equality theorems without distinct variables
Theorem	a9e 1632	At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1382 through ax-14 1451 and ax-17 1465, all axioms other than ax-9 1470 are believed to be theorems of free logic, although the system without ax-9 1470 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
|- E. x x = y
Theorem	a9ev 1633*	At least one individual exists. Weaker version of a9e 1632. (Contributed by NM, 3-Aug-2017.)
|- E. x x = y
Theorem	ax9o 1634	An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
|- ( A. x ( x = y -> A. x ph ) -> ph )
Theorem	equid 1635	Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms. This proof is similar to Tarski's and makes use of a dummy variable y. It also works in intuitionistic logic, unlike some other possible ways of proving this theorem. (Contributed by NM, 1-Apr-2005.)
|- x = x
Theorem	nfequid 1636	Bound-variable hypothesis builder for x = x. This theorem tells us that any variable, including x, is effectively not free in x = x, even though x is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)
|- F/ y x = x
Theorem	stdpc6 1637	One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1701.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)
|- A. x x = x
Theorem	equcomi 1638	Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> y = x )
Theorem	ax6evr 1639*	A commuted form of a9ev 1633. The naming reflects how axioms were numbered in the Metamath Proof Explorer as of 2020 (a numbering which we eventually plan to adopt here too, but until this happens everywhere only some theorems will have it). (Contributed by BJ, 7-Dec-2020.)
|- E. x y = x
Theorem	equcom 1640	Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
|- ( x = y <-> y = x )
Theorem	equcomd 1641	Deduction form of equcom 1640, symmetry of equality. For the versions for classes, see eqcom 2091 and eqcomd 2094. (Contributed by BJ, 6-Oct-2019.)
|- ( ph -> x = y )   =>    |- ( ph -> y = x )
Theorem	equcoms 1642	An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ph )   =>    |- ( y = x -> ph )
Theorem	equtr 1643	A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
|- ( x = y -> ( y = z -> x = z ) )
Theorem	equtrr 1644	A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)
|- ( x = y -> ( z = x -> z = y ) )
Theorem	equtr2 1645	A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ( x = z /\ y = z ) -> x = y )
Theorem	equequ1 1646	An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( x = z <-> y = z ) )
Theorem	equequ2 1647	An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( z = x <-> z = y ) )
Theorem	elequ1 1648	An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( x e. z <-> y e. z ) )
Theorem	elequ2 1649	An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( z e. x <-> z e. y ) )
Theorem	ax11i 1650	Inference that has ax-11 1443 (without A. y) as its conclusion and doesn't require ax-10 1442, ax-11 1443, or ax-12 1448 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( ps -> A. x ps )   =>    |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
1.3.9  Axioms ax-10 and ax-11
Theorem	ax10o 1651	Show that ax-10o 1652 can be derived from ax-10 1442. An open problem is whether this theorem can be derived from ax-10 1442 and the others when ax-11 1443 is replaced with ax-11o 1752. See theorem ax10 1653 for the rederivation of ax-10 1442 from ax10o 1651. Normally, ax10o 1651 should be used rather than ax-10o 1652, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)
|- ( A. x x = y -> ( A. x ph -> A. y ph ) )
Axiom	ax-10o 1652	Axiom ax-10o 1652 ("o" for "old") was the original version of ax-10 1442, before it was discovered (in May 2008) that the shorter ax-10 1442 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint). This axiom is redundant, as shown by theorem ax10o 1651. Normally, ax10o 1651 should be used rather than ax-10o 1652, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
|- ( A. x x = y -> ( A. x ph -> A. y ph ) )
Theorem	ax10 1653	Rederivation of ax-10 1442 from original version ax-10o 1652. See theorem ax10o 1651 for the derivation of ax-10o 1652 from ax-10 1442. This theorem should not be referenced in any proof. Instead, use ax-10 1442 above so that uses of ax-10 1442 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)
|- ( A. x x = y -> A. y y = x )
Theorem	hbae 1654	All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
|- ( A. x x = y -> A. z A. x x = y )
Theorem	nfae 1655	All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ z A. x x = y
Theorem	hbaes 1656	Rule that applies hbae 1654 to antecedent. (Contributed by NM, 5-Aug-1993.)
|- ( A. z A. x x = y -> ph )   =>    |- ( A. x x = y -> ph )
Theorem	hbnae 1657	All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.)
|- ( -. A. x x = y -> A. z -. A. x x = y )
Theorem	nfnae 1658	All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ z -. A. x x = y
Theorem	hbnaes 1659	Rule that applies hbnae 1657 to antecedent. (Contributed by NM, 5-Aug-1993.)
|- ( A. z -. A. x x = y -> ph )   =>    |- ( -. A. x x = y -> ph )
Theorem	naecoms 1660	A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
|- ( -. A. x x = y -> ph )   =>    |- ( -. A. y y = x -> ph )
Theorem	equs4 1661	Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)
|- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
Theorem	equsalh 1662	A useful equivalence related to substitution. New proofs should use equsal 1663 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ( x = y -> ph ) <-> ps )
Theorem	equsal 1663	A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ( x = y -> ph ) <-> ps )
Theorem	equsex 1664	A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ( x = y /\ ph ) <-> ps )
Theorem	equsexd 1665	Deduction form of equsex 1664. (Contributed by Jim Kingdon, 29-Dec-2017.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( E. x ( x = y /\ ps ) <-> ch ) )
Theorem	dral1 1666	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.)
|- ( A. x x = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( A. x ph <-> A. y ps ) )
Theorem	dral2 1667	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
|- ( A. x x = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( A. z ph <-> A. z ps ) )
Theorem	drex2 1668	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
|- ( A. x x = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( E. z ph <-> E. z ps ) )
Theorem	drnf1 1669	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
|- ( A. x x = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) )
Theorem	drnf2 1670	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
|- ( A. x x = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( F/ z ph <-> F/ z ps ) )
Theorem	spimth 1671	Closed theorem form of spim 1674. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.)
|- ( A. x ( ( ps -> A. x ps ) /\ ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) )
Theorem	spimt 1672	Closed theorem form of spim 1674. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.)
|- ( ( F/ x ps /\ A. x ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) )
Theorem	spimh 1673	Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1674 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 8-May-2008.) (New usage is discouraged.)
|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	spim 1674	Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1674 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
|- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	spimeh 1675	Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.)
|- ( ph -> A. x ph )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( ph -> E. x ps )
Theorem	spimed 1676	Deduction version of spime 1677. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)
|- ( ch -> F/ x ph )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( ch -> ( ph -> E. x ps ) )
Theorem	spime 1677	Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.)
|- F/ x ph   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( ph -> E. x ps )
Theorem	cbv3 1678	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> A. y ps )
Theorem	cbv3h 1679	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.)
|- ( ph -> A. y ph )   &    |- ( ps -> A. x ps )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> A. y ps )
Theorem	cbv1 1680	Rule used to change bound variables, using implicit substitution. Revised to format hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
|- F/ x ph   &    |- F/ y ph   &    |- ( ph -> F/ y ps )   &    |- ( ph -> F/ x ch )   &    |- ( ph -> ( x = y -> ( ps -> ch ) ) )   =>    |- ( ph -> ( A. x ps -> A. y ch ) )
Theorem	cbv1h 1681	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)
|- ( ph -> ( ps -> A. y ps ) )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( x = y -> ( ps -> ch ) ) )   =>    |- ( A. x A. y ph -> ( A. x ps -> A. y ch ) )
Theorem	cbv2h 1682	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps -> A. y ps ) )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) )
Theorem	cbv2 1683	Rule used to change bound variables, using implicit substitution. Revised to align format of hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
|- F/ x ph   &    |- F/ y ph   &    |- ( ph -> F/ y ps )   &    |- ( ph -> F/ x ch )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( A. x ps <-> A. y ch ) )
Theorem	cbvalh 1684	Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A. y ph )   &    |- ( ps -> A. x ps )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph <-> A. y ps )
Theorem	cbval 1685	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph <-> A. y ps )
Theorem	cbvexh 1686	Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.)
|- ( ph -> A. y ph )   &    |- ( ps -> A. x ps )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ph <-> E. y ps )
Theorem	cbvex 1687	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ph <-> E. y ps )
Theorem	chvar 1688	Implicit substitution of y for x into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	equvini 1689	A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require z to be distinct from x and y (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( x = y -> E. z ( x = z /\ z = y ) )
Theorem	equveli 1690	A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1689.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)
|- ( A. z ( z = x <-> z = y ) -> x = y )
Theorem	nfald 1691	If x is not free in ph, it is not free in A. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x A. y ps )
Theorem	nfexd 1692	If x is not free in ph, it is not free in E. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 7-Feb-2018.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x E. y ps )
1.3.10  Substitution (without distinct variables)
Syntax	wsb 1693	Extend wff definition to include proper substitution (read "the wff that results when y is properly substituted for x in wff ph"). (Contributed by NM, 24-Jan-2006.)
wff [ y / x ] ph
Definition	df-sb 1694	Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use [ y / x ] ph to mean "the wff that results when y is properly substituted for x in the wff ph." We can also use [ y / x ] ph in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 1706. Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, " ph ( y ) is the wff that results when y is properly substituted for x in ph ( x )." For example, if the original ph ( x ) is x = y, then ph ( y ) is y = y, from which we obtain that ph ( x ) is x = x. So what exactly does ph ( x ) mean? Curry's notation solves this problem. In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 1769, sbcom2 1912 and sbid2v 1921). Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1705 shows. We achieve this by having x free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 1916 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another alternate definition which uses a dummy variable is dfsb7a 1919. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1816 and sb6 1815. In classical logic, another possible definition is ( x = y /\ ph ) \/ A. x ( x = y -> ph ) but we do not have an intuitionistic proof that this is equivalent. There are no restrictions on any of the variables, including what variables may occur in wff ph. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
Theorem	sbimi 1695	Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
|- ( ph -> ps )   =>    |- ( [ y / x ] ph -> [ y / x ] ps )
Theorem	sbbii 1696	Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   =>    |- ( [ y / x ] ph <-> [ y / x ] ps )
Theorem	sb1 1697	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
Theorem	sb2 1698	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
Theorem	sbequ1 1699	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph -> [ y / x ] ph ) )
Theorem	sbequ2 1700	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( [ y / x ] ph -> ph ) )