P. 19 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 1801-1900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cbv2h 1801	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 1802	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 1803*	Rule used to change bound variables, using implicit substitution. Version of cbv2 1802 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by GG, 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 1804*	Rule used to change bound variables, using implicit substitution. Version of cbval 1807 with a disjoint variable condition. See cbvalvw 1975 for a version with two disjoint variable conditions, and cbvalv 1973 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 1805*	Rule used to change bound variables, using implicit substitution. Version of cbvex 1809 with a disjoint variable condition. See cbvexvw 1976 for a version with two disjoint variable conditions, and cbvexv 1974 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 1806	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 1807	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 1808	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 1809	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 1810	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 1811	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 1812	A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1811.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)
|- ( A. z ( z = x <-> z = y ) -> x = y )
Theorem	nfald 1813	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 1814	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 1815	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 1816	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 1828. 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 1893, sbcom2 2047 and sbid2v 2056). Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1827 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 2051 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 2054. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1942 and sb6 1941. 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 1817	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 1818	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 1819	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
Theorem	sb2 1820	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
Theorem	sbequ1 1821	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph -> [ y / x ] ph ) )
Theorem	sbequ2 1822	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( [ y / x ] ph -> ph ) )
Theorem	stdpc7 1823	One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1755.) 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 1824	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph <-> [ y / x ] ph ) )
Theorem	sbequ12r 1825	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 1826	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )
Theorem	sbid 1827	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 1828	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 1829	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 1830	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 1831	Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
|- ( [ y / x ] A. x ph <-> A. x ph )
Theorem	sb6x 1832	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 1833	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 1834	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 1835	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 1836	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 1837	A substitution into a theorem remains true. (See chvar 1810 and chvarv 1997 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 1838	Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
|- [ y / x ] x = y
Theorem	equsb2 1839	Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
|- [ y / x ] y = x
Theorem	sbiedh 1840	Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1843). New proofs should use sbied 1841 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 1841	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1844). (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 1842*	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1844). (Contributed by NM, 7-Jan-2017.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( [ y / x ] ps <-> ch ) )
Theorem	sbieh 1843	Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1844 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 1844	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 1845*	Conversion of implicit substitution to explicit substitution. Version of sbie 1844 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 1846*	An equivalence related to implicit substitution. Version of equsal 1779 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 1847	A property related to substitution that unlike equs5 1882 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 1848	A property related to substitution that unlike equs5 1882 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 1849	Analogue to ax-11 1559 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 1850	Quantifier Substitution for existential quantifiers. Analogue to ax10o 1767 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 1851	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 1852	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 1853	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 1854	A version of sb4 1885 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 1855	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 ) )
Theorem	sb6f 1856	Equivalence for substitution when y is not free in ph. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.)
|- ( ph -> A. y ph )   =>    |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
Theorem	sb5f 1857	Equivalence for substitution when y is not free in ph. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.)
|- ( ph -> A. y ph )   =>    |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
Theorem	sb4e 1858	One direction of a simplified definition of substitution that unlike sb4 1885 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
|- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) )
Theorem	hbsb2a 1859	Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
|- ( [ y / x ] A. y ph -> A. x [ y / x ] ph )
Theorem	hbsb2e 1860	Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
|- ( [ y / x ] ph -> A. x [ y / x ] E. y ph )
Theorem	hbsb3 1861	If y is not free in ph, x is not free in [ y / x ] ph. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. y ph )   =>    |- ( [ y / x ] ph -> A. x [ y / x ] ph )
Theorem	nfs1 1862	If y is not free in ph, x is not free in [ y / x ] ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ y ph   =>    |- F/ x [ y / x ] ph
Theorem	sbcof2 1863	Version of sbco 2028 where x is not free in ph. (Contributed by Jim Kingdon, 28-Dec-2017.)
|- ( ph -> A. x ph )   =>    |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )
1.4  Predicate calculus with distinct variables
1.4.1  Derive the axiom of distinct variables ax-16
Theorem	spimv 1864*	A version of spim 1791 with a distinct variable requirement instead of a bound-variable hypothesis. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	aev 1865*	A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1867. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
|- ( A. x x = y -> A. z w = v )
Theorem	ax16 1866*	Theorem showing that ax-16 1867 is redundant if ax-17 1579 is included in the axiom system. The important part of the proof is provided by aev 1865. See ax16ALT 1912 for an alternate proof that does not require ax-10 1558 or ax12 1565. This theorem should not be referenced in any proof. Instead, use ax-16 1867 below so that theorems needing ax-16 1867 can be more easily identified. (Contributed by NM, 8-Nov-2006.)
|- ( A. x x = y -> ( ph -> A. x ph ) )
Axiom	ax-16 1867*	Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1579 to be a metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. It is a somewhat bizarre axiom since the antecedent is always false in set theory, but nonetheless it is technically necessary as you can see from its uses. This axiom is redundant if we include ax-17 1579; see Theorem ax16 1866. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax16 1866. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
|- ( A. x x = y -> ( ph -> A. x ph ) )
Theorem	dveeq2 1868*	Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
|- ( -. A. x x = y -> ( z = y -> A. x z = y ) )
Theorem	dveeq2or 1869*	Quantifier introduction when one pair of variables is distinct. Like dveeq2 1868 but connecting A. x x = y by a disjunction rather than negation and implication makes the theorem stronger in intuitionistic logic. (Contributed by Jim Kingdon, 1-Feb-2018.)
|- ( A. x x = y \/ F/ x z = y )
Theorem	dvelimfALT2 1870*	Proof of dvelimf 2075 using dveeq2 1868 (shown as the last hypothesis) instead of ax12 1565. This shows that ax12 1565 could be replaced by dveeq2 1868 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.)
|- ( ph -> A. x ph )   &    |- ( ps -> A. z ps )   &    |- ( z = y -> ( ph <-> ps ) )   &    |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )   =>    |- ( -. A. x x = y -> ( ps -> A. x ps ) )
Theorem	nd5 1871*	A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.)
|- ( -. A. y y = x -> ( z = y -> A. x z = y ) )
Theorem	exlimdv 1872*	Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x ps -> ch ) )
Theorem	ax11v2 1873*	Recovery of ax11o 1875 from ax11v 1880 without using ax-11 1559. The hypothesis is even weaker than ax11v 1880, with z both distinct from x and not occurring in ph. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1875. (Contributed by NM, 2-Feb-2007.)
|- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )   =>    |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
Theorem	ax11a2 1874*	Derive ax-11o 1876 from a hypothesis in the form of ax-11 1559. The hypothesis is even weaker than ax-11 1559, with z both distinct from x and not occurring in ph. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1875. (Contributed by NM, 2-Feb-2007.)
|- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) )   =>    |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
1.4.2  Derive the obsolete axiom of variable substitution ax-11o
Theorem	ax11o 1875	Derivation of set.mm's original ax-11o 1876 from the shorter ax-11 1559 that has replaced it. An open problem is whether this theorem can be proved without relying on ax-16 1867 or ax-17 1579. Normally, ax11o 1875 should be used rather than ax-11o 1876, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)
|- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
Axiom	ax-11o 1876	Axiom ax-11o 1876 ("o" for "old") was the original version of ax-11 1559, before it was discovered (in Jan. 2007) that the shorter ax-11 1559 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. To understand this theorem more easily, think of " -. A. x x = y ->..." as informally meaning "if x and y are distinct variables, then..." The antecedent becomes false if the same variable is substituted for x and y, ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form -. A. x x = y a "distinctor." This axiom is redundant, as shown by Theorem ax11o 1875. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax11o 1875. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
|- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
1.4.3  More theorems related to ax-11 and substitution
Theorem	albidv 1877*	Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps <-> A. x ch ) )
Theorem	exbidv 1878*	Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x ps <-> E. x ch ) )
Theorem	ax11b 1879	A bidirectional version of ax-11o 1876. (Contributed by NM, 30-Jun-2006.)
|- ( ( -. A. x x = y /\ x = y ) -> ( ph <-> A. x ( x = y -> ph ) ) )
Theorem	ax11v 1880*	This is a version of ax-11o 1876 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.)
|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
Theorem	ax11ev 1881*	Analogue to ax11v 1880 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
|- ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) )
Theorem	equs5 1882	Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
|- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
Theorem	equs5or 1883	Lemma used in proofs of substitution properties. Like equs5 1882 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.)
|- ( A. x x = y \/ ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
Theorem	sb3 1884	One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
|- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> [ y / x ] ph ) )
Theorem	sb4 1885	One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
|- ( -. A. x x = y -> ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )
Theorem	sb4or 1886	One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1885 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)
|- ( A. x x = y \/ A. x ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )
Theorem	sb4b 1887	Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
|- ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )
Theorem	sb4bor 1888	Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.)
|- ( A. x x = y \/ A. x ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )
Theorem	hbsb2 1889	Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( -. A. x x = y -> ( [ y / x ] ph -> A. x [ y / x ] ph ) )
Theorem	nfsb2or 1890	Bound-variable hypothesis builder for substitution. Similar to hbsb2 1889 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.)
|- ( A. x x = y \/ F/ x [ y / x ] ph )
Theorem	sbequilem 1891	Propositional logic lemma used in the sbequi 1892 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
|- ( ph \/ ( ps -> ( ch -> th ) ) )   &    |- ( ta \/ ( ps -> ( th -> et ) ) )   =>    |- ( ph \/ ( ta \/ ( ps -> ( ch -> et ) ) ) )
Theorem	sbequi 1892	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.)
|- ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) )
Theorem	sbequ 1893	An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) )
Theorem	drsb2 1894	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 -> ( [ x / z ] ph <-> [ y / z ] ph ) )
Theorem	spsbe 1895	A specialization theorem, mostly the same as Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 29-Dec-2017.)
|- ( [ y / x ] ph -> E. x ph )
Theorem	spsbim 1896	Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
|- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
Theorem	spsbbi 1897	Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
|- ( A. x ( ph <-> ps ) -> ( [ y / x ] ph <-> [ y / x ] ps ) )
Theorem	sbbidh 1898	Deduction substituting both sides of a biconditional. New proofs should use sbbid 1899 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( [ y / x ] ps <-> [ y / x ] ch ) )
Theorem	sbbid 1899	Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( [ y / x ] ps <-> [ y / x ] ch ) )
Theorem	sbequ8 1900	Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) (Proof revised by Jim Kingdon, 20-Jan-2018.)
|- ( [ y / x ] ph <-> [ y / x ] ( x = y -> ph ) )