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	cbvexh 1801	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 1802	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 1803	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 1804	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 1805	A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1804.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)
|- ( A. z ( z = x <-> z = y ) -> x = y )
Theorem	nfald 1806	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 1807	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 1808	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 1809	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 1821. 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 1886, sbcom2 2038 and sbid2v 2047). Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1820 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 2042 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 2045. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1934 and sb6 1933. 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 1810	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 1811	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 1812	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
Theorem	sb2 1813	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
Theorem	sbequ1 1814	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph -> [ y / x ] ph ) )
Theorem	sbequ2 1815	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( [ y / x ] ph -> ph ) )
Theorem	stdpc7 1816	One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1749.) 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 1817	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph <-> [ y / x ] ph ) )
Theorem	sbequ12r 1818	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 1819	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )
Theorem	sbid 1820	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 1821	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 1822	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 1823	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 1824	Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
|- ( [ y / x ] A. x ph <-> A. x ph )
Theorem	sb6x 1825	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 1826	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 1827	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 1828	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 1829	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 1830	A substitution into a theorem remains true. (See chvar 1803 and chvarv 1988 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 1831	Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
|- [ y / x ] x = y
Theorem	equsb2 1832	Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
|- [ y / x ] y = x
Theorem	sbiedh 1833	Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1836). New proofs should use sbied 1834 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 1834	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1837). (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 1835*	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1837). (Contributed by NM, 7-Jan-2017.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( [ y / x ] ps <-> ch ) )
Theorem	sbieh 1836	Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1837 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 1837	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 1838*	Conversion of implicit substitution to explicit substitution. Version of sbie 1837 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 1839*	An equivalence related to implicit substitution. Version of equsal 1773 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 1840	A property related to substitution that unlike equs5 1875 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 1841	A property related to substitution that unlike equs5 1875 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 1842	Analogue to ax-11 1552 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 1843	Quantifier Substitution for existential quantifiers. Analogue to ax10o 1761 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 1844	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 1845	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 1846	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 1847	A version of sb4 1878 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 1848	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 1849	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 1850	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 1851	One direction of a simplified definition of substitution that unlike sb4 1878 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
|- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) )
Theorem	hbsb2a 1852	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 1853	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 1854	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 1855	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 1856	Version of sbco 2019 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 1857*	A version of spim 1784 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 1858*	A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1860. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
|- ( A. x x = y -> A. z w = v )
Theorem	ax16 1859*	Theorem showing that ax-16 1860 is redundant if ax-17 1572 is included in the axiom system. The important part of the proof is provided by aev 1858. See ax16ALT 1905 for an alternate proof that does not require ax-10 1551 or ax12 1558. This theorem should not be referenced in any proof. Instead, use ax-16 1860 below so that theorems needing ax-16 1860 can be more easily identified. (Contributed by NM, 8-Nov-2006.)
|- ( A. x x = y -> ( ph -> A. x ph ) )
Axiom	ax-16 1860*	Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1572 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 1572; see Theorem ax16 1859. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax16 1859. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
|- ( A. x x = y -> ( ph -> A. x ph ) )
Theorem	dveeq2 1861*	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 1862*	Quantifier introduction when one pair of variables is distinct. Like dveeq2 1861 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 1863*	Proof of dvelimf 2066 using dveeq2 1861 (shown as the last hypothesis) instead of ax12 1558. This shows that ax12 1558 could be replaced by dveeq2 1861 (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 1864*	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 1865*	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 1866*	Recovery of ax11o 1868 from ax11v 1873 without using ax-11 1552. The hypothesis is even weaker than ax11v 1873, 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 1868. (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 1867*	Derive ax-11o 1869 from a hypothesis in the form of ax-11 1552. The hypothesis is even weaker than ax-11 1552, 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 1868. (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 1868	Derivation of set.mm's original ax-11o 1869 from the shorter ax-11 1552 that has replaced it. An open problem is whether this theorem can be proved without relying on ax-16 1860 or ax-17 1572. Normally, ax11o 1868 should be used rather than ax-11o 1869, 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 1869	Axiom ax-11o 1869 ("o" for "old") was the original version of ax-11 1552, before it was discovered (in Jan. 2007) that the shorter ax-11 1552 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 1868. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax11o 1868. (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 1870*	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 1871*	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 1872	A bidirectional version of ax-11o 1869. (Contributed by NM, 30-Jun-2006.)
|- ( ( -. A. x x = y /\ x = y ) -> ( ph <-> A. x ( x = y -> ph ) ) )
Theorem	ax11v 1873*	This is a version of ax-11o 1869 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 1874*	Analogue to ax11v 1873 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
|- ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) )
Theorem	equs5 1875	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 1876	Lemma used in proofs of substitution properties. Like equs5 1875 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 1877	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 1878	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 1879	One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1878 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 1880	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 1881	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 1882	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 1883	Bound-variable hypothesis builder for substitution. Similar to hbsb2 1882 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 1884	Propositional logic lemma used in the sbequi 1885 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
|- ( ph \/ ( ps -> ( ch -> th ) ) )   &    |- ( ta \/ ( ps -> ( th -> et ) ) )   =>    |- ( ph \/ ( ta \/ ( ps -> ( ch -> et ) ) ) )
Theorem	sbequi 1885	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 1886	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 1887	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 1888	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 1889	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 1890	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 1891	Deduction substituting both sides of a biconditional. New proofs should use sbbid 1892 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 1892	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 1893	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 ) )
Theorem	sbft 1894	Substitution has no effect on a nonfree variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)
|- ( F/ x ph -> ( [ y / x ] ph <-> ph ) )
Theorem	sbid2h 1895	An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   =>    |- ( [ y / x ] [ x / y ] ph <-> ph )
Theorem	sbid2 1896	An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
|- F/ x ph   =>    |- ( [ y / x ] [ x / y ] ph <-> ph )
Theorem	sbidm 1897	An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
|- ( [ y / x ] [ y / x ] ph <-> [ y / x ] ph )
Theorem	sb5rf 1898	Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A. y ph )   =>    |- ( ph <-> E. y ( y = x /\ [ y / x ] ph ) )
Theorem	sb6rf 1899	Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A. y ph )   =>    |- ( ph <-> A. y ( y = x -> [ y / x ] ph ) )
Theorem	sb8h 1900	Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
|- ( ph -> A. y ph )   =>    |- ( A. x ph <-> A. y [ y / x ] ph )