P. 18 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 1701-1800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	stdpc6 1701	One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1768.) 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 1702	Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> y = x )
Theorem	ax6evr 1703*	A commuted form of a9ev 1695. 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 1704	Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
|- ( x = y <-> y = x )
Theorem	equcomd 1705	Deduction form of equcom 1704, symmetry of equality. For the versions for classes, see eqcom 2177 and eqcomd 2181. (Contributed by BJ, 6-Oct-2019.)
|- ( ph -> x = y )   =>    |- ( ph -> y = x )
Theorem	equcoms 1706	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 1707	A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
|- ( x = y -> ( y = z -> x = z ) )
Theorem	equtrr 1708	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 1709	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 1710	An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( x = z <-> y = z ) )
Theorem	equequ2 1711	An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( z = x <-> z = y ) )
Theorem	ax11i 1712	Inference that has ax-11 1504 (without A. y) as its conclusion and does not require ax-10 1503, ax-11 1504, or ax12 1510 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 1713	Show that ax-10o 1714 can be derived from ax-10 1503. An open problem is whether this theorem can be derived from ax-10 1503 and the others when ax-11 1504 is replaced with ax-11o 1821. See Theorem ax10 1715 for the rederivation of ax-10 1503 from ax10o 1713. Normally, ax10o 1713 should be used rather than ax-10o 1714, 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 1714	Axiom ax-10o 1714 ("o" for "old") was the original version of ax-10 1503, before it was discovered (in May 2008) that the shorter ax-10 1503 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 1713. Normally, ax10o 1713 should be used rather than ax-10o 1714, 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 1715	Rederivation of ax-10 1503 from original version ax-10o 1714. See Theorem ax10o 1713 for the derivation of ax-10o 1714 from ax-10 1503. This theorem should not be referenced in any proof. Instead, use ax-10 1503 above so that uses of ax-10 1503 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 1716	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 1717	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 1718	Rule that applies hbae 1716 to antecedent. (Contributed by NM, 5-Aug-1993.)
|- ( A. z A. x x = y -> ph )   =>    |- ( A. x x = y -> ph )
Theorem	hbnae 1719	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 1720	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 1721	Rule that applies hbnae 1719 to antecedent. (Contributed by NM, 5-Aug-1993.)
|- ( A. z -. A. x x = y -> ph )   =>    |- ( -. A. x x = y -> ph )
Theorem	naecoms 1722	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 1723	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 1724	A useful equivalence related to substitution. New proofs should use equsal 1725 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 1725	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 1726	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 1727	Deduction form of equsex 1726. (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 1728	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 1729	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 1730	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 1731	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 1732	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 1733	Closed theorem form of spim 1736. (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 1734	Closed theorem form of spim 1736. (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 1735	Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1736 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 1736	Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1736 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 1737	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 1738	Deduction version of spime 1739. (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 1739	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 1740	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because proofs are encouraged to use the weaker cbv3v 1742 if possible. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> A. y ps )
Theorem	cbv3h 1741	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	cbv3v 1742*	Rule used to change bound variables, using implicit substitution. Version of cbv3 1740 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> A. y ps )
Theorem	cbv1 1743	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 1744	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	cbv1v 1745*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 16-Jun-2019.)
|- 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	cbv2h 1746	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 1747	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	cbv2w 1748*	Rule used to change bound variables, using implicit substitution. Version of cbv2 1747 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.)
|- 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	cbvalv1 1749*	Rule used to change bound variables, using implicit substitution. Version of cbval 1752 with a disjoint variable condition. See cbvalvw 1917 for a version with two disjoint variable conditions, and cbvalv 1915 for another variant. (Contributed by NM, 13-May-1993.) (Revised by BJ, 31-May-2019.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph <-> A. y ps )
Theorem	cbvexv1 1750*	Rule used to change bound variables, using implicit substitution. Version of cbvex 1754 with a disjoint variable condition. See cbvexvw 1918 for a version with two disjoint variable conditions, and cbvexv 1916 for another variant. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 31-May-2019.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ph <-> E. y ps )
Theorem	cbvalh 1751	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 1752	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 1753	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 1754	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 1755	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 1756	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 1757	A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1756.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)
|- ( A. z ( z = x <-> z = y ) -> x = y )
Theorem	nfald 1758	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 1759	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 1760	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 1761	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 1773. 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 1838, sbcom2 1985 and sbid2v 1994). Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1772 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 1989 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 1992. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1885 and sb6 1884. 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 1762	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 1763	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 1764	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
Theorem	sb2 1765	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
Theorem	sbequ1 1766	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph -> [ y / x ] ph ) )
Theorem	sbequ2 1767	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( [ y / x ] ph -> ph ) )
Theorem	stdpc7 1768	One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1701.) Translated to traditional notation, it can be read: " x = y -> ( ph ( x, x ) -> ph ( x, y ) ), provided that y is free for x in ph ( x, y )". Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
|- ( x = y -> ( [ x / y ] ph -> ph ) )
Theorem	sbequ12 1769	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph <-> [ y / x ] ph ) )
Theorem	sbequ12r 1770	An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
|- ( x = y -> ( [ x / y ] ph <-> ph ) )
Theorem	sbequ12a 1771	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )
Theorem	sbid 1772	An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.)
|- ( [ x / x ] ph <-> ph )
Theorem	stdpc4 1773	The specialization axiom of standard predicate calculus. It states that if a statement ph holds for all x, then it also holds for the specific case of y (properly) substituted for x. Translated to traditional notation, it can be read: " A. x ph ( x ) -> ph ( y ), provided that y is free for x in ph ( x )". Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ph -> [ y / x ] ph )
Theorem	sbh 1774	Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.)
|- ( ph -> A. x ph )   =>    |- ( [ y / x ] ph <-> ph )
Theorem	sbf 1775	Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
|- F/ x ph   =>    |- ( [ y / x ] ph <-> ph )
Theorem	sbf2 1776	Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
|- ( [ y / x ] A. x ph <-> A. x ph )
Theorem	sb6x 1777	Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|- ( ph -> A. x ph )   =>    |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
Theorem	nfs1f 1778	If x is not free in ph, it is not free in [ y / x ] ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph   =>    |- F/ x [ y / x ] ph
Theorem	hbs1f 1779	If x is not free in ph, it is not free in [ y / x ] ph. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A. x ph )   =>    |- ( [ y / x ] ph -> A. x [ y / x ] ph )
Theorem	sbequ5 1780	Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-Dec-2004.)
|- ( [ w / z ] A. x x = y <-> A. x x = y )
Theorem	sbequ6 1781	Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 14-May-2005.)
|- ( [ w / z ] -. A. x x = y <-> -. A. x x = y )
Theorem	sbt 1782	A substitution into a theorem remains true. (See chvar 1755 and chvarv 1935 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ph   =>    |- [ y / x ] ph
Theorem	equsb1 1783	Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
|- [ y / x ] x = y
Theorem	equsb2 1784	Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
|- [ y / x ] y = x
Theorem	sbiedh 1785	Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1788). New proofs should use sbied 1786 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( [ y / x ] ps <-> ch ) )
Theorem	sbied 1786	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1789). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)
|- F/ x ph   &    |- ( ph -> F/ x ch )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( [ y / x ] ps <-> ch ) )
Theorem	sbiedv 1787*	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1789). (Contributed by NM, 7-Jan-2017.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( [ y / x ] ps <-> ch ) )
Theorem	sbieh 1788	Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1789 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.)
|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( [ y / x ] ph <-> ps )
Theorem	sbie 1789	Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Revised by Wolf Lammen, 30-Apr-2018.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( [ y / x ] ph <-> ps )
Theorem	sbiev 1790*	Conversion of implicit substitution to explicit substitution. Version of sbie 1789 with a disjoint variable condition. (Contributed by Wolf Lammen, 18-Jan-2023.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( [ y / x ] ph <-> ps )
Theorem	equsalv 1791*	An equivalence related to implicit substitution. Version of equsal 1725 with a disjoint variable condition. (Contributed by NM, 2-Jun-1993.) (Revised by BJ, 31-May-2019.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ( x = y -> ph ) <-> ps )
1.3.11  Theorems using axiom ax-11
Theorem	equs5a 1792	A property related to substitution that unlike equs5 1827 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
|- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )
Theorem	equs5e 1793	A property related to substitution that unlike equs5 1827 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.)
|- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )
Theorem	ax11e 1794	Analogue to ax-11 1504 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.)
|- ( x = y -> ( E. x ( x = y /\ ph ) -> E. y ph ) )
Theorem	ax10oe 1795	Quantifier Substitution for existential quantifiers. Analogue to ax10o 1713 but for E. rather than A.. (Contributed by Jim Kingdon, 21-Dec-2017.)
|- ( A. x x = y -> ( E. x ps -> E. y ps ) )
Theorem	drex1 1796	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.) (Revised by NM, 3-Feb-2015.)
|- ( A. x x = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( E. x ph <-> E. y ps ) )
Theorem	drsb1 1797	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, 5-Aug-1993.)
|- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) )
Theorem	exdistrfor 1798	Distribution of existential quantifiers, with a bound-variable hypothesis saying that y is not free in ph, but x can be free in ph (and there is no distinct variable condition on x and y). (Contributed by Jim Kingdon, 25-Feb-2018.)
|- ( A. x x = y \/ A. x F/ y ph )   =>    |- ( E. x E. y ( ph /\ ps ) -> E. x ( ph /\ E. y ps ) )
Theorem	sb4a 1799	A version of sb4 1830 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
|- ( [ y / x ] A. y ph -> A. x ( x = y -> ph ) )
Theorem	equs45f 1800	Two ways of expressing substitution when y is not free in ph. (Contributed by NM, 25-Apr-2008.)
|- ( ph -> A. y ph )   =>    |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )