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	cbvalh 1801	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 1802	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 1803	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 1804	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 1805	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 1806	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 1807	A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1806.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)
|- ( A. z ( z = x <-> z = y ) -> x = y )
Theorem	nfald 1808	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 1809	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 1810	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 1811	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 1823. 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 1888, sbcom2 2040 and sbid2v 2049). Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1822 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 2044 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 2047. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1936 and sb6 1935. 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 1812	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 1813	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 1814	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
Theorem	sb2 1815	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
Theorem	sbequ1 1816	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph -> [ y / x ] ph ) )
Theorem	sbequ2 1817	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( [ y / x ] ph -> ph ) )
Theorem	stdpc7 1818	One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1751.) 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 1819	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph <-> [ y / x ] ph ) )
Theorem	sbequ12r 1820	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 1821	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )
Theorem	sbid 1822	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 1823	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 1824	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 1825	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 1826	Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
|- ( [ y / x ] A. x ph <-> A. x ph )
Theorem	sb6x 1827	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 1828	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 1829	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 1830	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 1831	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 1832	A substitution into a theorem remains true. (See chvar 1805 and chvarv 1990 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 1833	Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
|- [ y / x ] x = y
Theorem	equsb2 1834	Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
|- [ y / x ] y = x
Theorem	sbiedh 1835	Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1838). New proofs should use sbied 1836 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 1836	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1839). (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 1837*	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1839). (Contributed by NM, 7-Jan-2017.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( [ y / x ] ps <-> ch ) )
Theorem	sbieh 1838	Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1839 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 1839	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 1840*	Conversion of implicit substitution to explicit substitution. Version of sbie 1839 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 1841*	An equivalence related to implicit substitution. Version of equsal 1775 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 1842	A property related to substitution that unlike equs5 1877 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 1843	A property related to substitution that unlike equs5 1877 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 1844	Analogue to ax-11 1554 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 1845	Quantifier Substitution for existential quantifiers. Analogue to ax10o 1763 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 1846	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 1847	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 1848	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 1849	A version of sb4 1880 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 1850	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 1851	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 1852	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 1853	One direction of a simplified definition of substitution that unlike sb4 1880 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
|- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) )
Theorem	hbsb2a 1854	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 1855	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 1856	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 1857	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 1858	Version of sbco 2021 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 1859*	A version of spim 1786 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 1860*	A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1862. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
|- ( A. x x = y -> A. z w = v )
Theorem	ax16 1861*	Theorem showing that ax-16 1862 is redundant if ax-17 1574 is included in the axiom system. The important part of the proof is provided by aev 1860. See ax16ALT 1907 for an alternate proof that does not require ax-10 1553 or ax12 1560. This theorem should not be referenced in any proof. Instead, use ax-16 1862 below so that theorems needing ax-16 1862 can be more easily identified. (Contributed by NM, 8-Nov-2006.)
|- ( A. x x = y -> ( ph -> A. x ph ) )
Axiom	ax-16 1862*	Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1574 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 1574; see Theorem ax16 1861. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax16 1861. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
|- ( A. x x = y -> ( ph -> A. x ph ) )
Theorem	dveeq2 1863*	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 1864*	Quantifier introduction when one pair of variables is distinct. Like dveeq2 1863 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 1865*	Proof of dvelimf 2068 using dveeq2 1863 (shown as the last hypothesis) instead of ax12 1560. This shows that ax12 1560 could be replaced by dveeq2 1863 (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 1866*	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 1867*	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 1868*	Recovery of ax11o 1870 from ax11v 1875 without using ax-11 1554. The hypothesis is even weaker than ax11v 1875, 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 1870. (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 1869*	Derive ax-11o 1871 from a hypothesis in the form of ax-11 1554. The hypothesis is even weaker than ax-11 1554, 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 1870. (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 1870	Derivation of set.mm's original ax-11o 1871 from the shorter ax-11 1554 that has replaced it. An open problem is whether this theorem can be proved without relying on ax-16 1862 or ax-17 1574. Normally, ax11o 1870 should be used rather than ax-11o 1871, 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 1871	Axiom ax-11o 1871 ("o" for "old") was the original version of ax-11 1554, before it was discovered (in Jan. 2007) that the shorter ax-11 1554 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 1870. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax11o 1870. (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 1872*	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 1873*	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 1874	A bidirectional version of ax-11o 1871. (Contributed by NM, 30-Jun-2006.)
|- ( ( -. A. x x = y /\ x = y ) -> ( ph <-> A. x ( x = y -> ph ) ) )
Theorem	ax11v 1875*	This is a version of ax-11o 1871 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 1876*	Analogue to ax11v 1875 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
|- ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) )
Theorem	equs5 1877	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 1878	Lemma used in proofs of substitution properties. Like equs5 1877 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 1879	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 1880	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 1881	One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1880 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 1882	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 1883	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 1884	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 1885	Bound-variable hypothesis builder for substitution. Similar to hbsb2 1884 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 1886	Propositional logic lemma used in the sbequi 1887 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
|- ( ph \/ ( ps -> ( ch -> th ) ) )   &    |- ( ta \/ ( ps -> ( th -> et ) ) )   =>    |- ( ph \/ ( ta \/ ( ps -> ( ch -> et ) ) ) )
Theorem	sbequi 1887	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 1888	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 1889	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 1890	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 1891	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 1892	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 1893	Deduction substituting both sides of a biconditional. New proofs should use sbbid 1894 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 1894	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 1895	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 1896	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 1897	An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   =>    |- ( [ y / x ] [ x / y ] ph <-> ph )
Theorem	sbid2 1898	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 1899	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 1900	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 ) )