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	nfald 1701	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 1702	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 1703	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 1704	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 1716. 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 1779, sbcom2 1923 and sbid2v 1932). Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1715 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 1927 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 1930. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1826 and sb6 1825. 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 1705	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 1706	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 1707	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
Theorem	sb2 1708	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
Theorem	sbequ1 1709	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph -> [ y / x ] ph ) )
Theorem	sbequ2 1710	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( [ y / x ] ph -> ph ) )
Theorem	stdpc7 1711	One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1647.) 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 1712	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph <-> [ y / x ] ph ) )
Theorem	sbequ12r 1713	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 1714	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )
Theorem	sbid 1715	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 1716	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 1717	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 1718	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 1719	Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
|- ( [ y / x ] A. x ph <-> A. x ph )
Theorem	sb6x 1720	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 1721	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 1722	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 1723	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 1724	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 1725	A substitution into a theorem remains true. (See chvar 1698 and chvarv 1872 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 1726	Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
|- [ y / x ] x = y
Theorem	equsb2 1727	Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
|- [ y / x ] y = x
Theorem	sbiedh 1728	Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1731). New proofs should use sbied 1729 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 1729	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1732). (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 1730*	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1732). (Contributed by NM, 7-Jan-2017.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( [ y / x ] ps <-> ch ) )
Theorem	sbieh 1731	Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1732 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 1732	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 )
1.3.11  Theorems using axiom ax-11
Theorem	equs5a 1733	A property related to substitution that unlike equs5 1768 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 1734	A property related to substitution that unlike equs5 1768 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 1735	Analogue to ax-11 1452 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 1736	Quantifier Substitution for existential quantifiers. Analogue to ax10o 1661 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 1737	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 1738	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 1739	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 1740	A version of sb4 1771 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 1741	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 1742	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 1743	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 1744	One direction of a simplified definition of substitution that unlike sb4 1771 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
|- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) )
Theorem	hbsb2a 1745	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 1746	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 1747	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 1748	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 1749	Version of sbco 1902 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 1750*	A version of spim 1684 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 1751*	A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1753. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
|- ( A. x x = y -> A. z w = v )
Theorem	ax16 1752*	Theorem showing that ax-16 1753 is redundant if ax-17 1474 is included in the axiom system. The important part of the proof is provided by aev 1751. See ax16ALT 1798 for an alternate proof that does not require ax-10 1451 or ax-12 1457. This theorem should not be referenced in any proof. Instead, use ax-16 1753 below so that theorems needing ax-16 1753 can be more easily identified. (Contributed by NM, 8-Nov-2006.)
|- ( A. x x = y -> ( ph -> A. x ph ) )
Axiom	ax-16 1753*	Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1474 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 1474; see theorem ax16 1752. This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1752. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
|- ( A. x x = y -> ( ph -> A. x ph ) )
Theorem	dveeq2 1754*	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 1755*	Quantifier introduction when one pair of variables is distinct. Like dveeq2 1754 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 1756*	Proof of dvelimf 1951 using dveeq2 1754 (shown as the last hypothesis) instead of ax-12 1457. This shows that ax-12 1457 could be replaced by dveeq2 1754 (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 1757*	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 1758*	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 1759*	Recovery of ax11o 1761 from ax11v 1766 without using ax-11 1452. The hypothesis is even weaker than ax11v 1766, 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 1761. (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 1760*	Derive ax-11o 1762 from a hypothesis in the form of ax-11 1452. The hypothesis is even weaker than ax-11 1452, 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 1761. (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 1761	Derivation of set.mm's original ax-11o 1762 from the shorter ax-11 1452 that has replaced it. An open problem is whether this theorem can be proved without relying on ax-16 1753 or ax-17 1474. Normally, ax11o 1761 should be used rather than ax-11o 1762, 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 1762	Axiom ax-11o 1762 ("o" for "old") was the original version of ax-11 1452, before it was discovered (in Jan. 2007) that the shorter ax-11 1452 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 1761. This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1761. (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 1763*	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 1764*	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 1765	A bidirectional version of ax-11o 1762. (Contributed by NM, 30-Jun-2006.)
|- ( ( -. A. x x = y /\ x = y ) -> ( ph <-> A. x ( x = y -> ph ) ) )
Theorem	ax11v 1766*	This is a version of ax-11o 1762 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 1767*	Analogue to ax11v 1766 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
|- ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) )
Theorem	equs5 1768	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 1769	Lemma used in proofs of substitution properties. Like equs5 1768 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 1770	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 1771	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 1772	One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1771 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 1773	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 1774	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 1775	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 1776	Bound-variable hypothesis builder for substitution. Similar to hbsb2 1775 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 1777	Propositional logic lemma used in the sbequi 1778 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
|- ( ph \/ ( ps -> ( ch -> th ) ) )   &    |- ( ta \/ ( ps -> ( th -> et ) ) )   =>    |- ( ph \/ ( ta \/ ( ps -> ( ch -> et ) ) ) )
Theorem	sbequi 1778	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 1779	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 1780	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 1781	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 1782	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 1783	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 1784	Deduction substituting both sides of a biconditional. New proofs should use sbbid 1785 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 1785	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 1786	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 1787	Substitution has no effect on a non-free 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 1788	An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   =>    |- ( [ y / x ] [ x / y ] ph <-> ph )
Theorem	sbid2 1789	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 1790	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 1791	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 1792	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 1793	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 )
Theorem	sb8eh 1794	Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.)
|- ( ph -> A. y ph )   =>    |- ( E. x ph <-> E. y [ y / x ] ph )
Theorem	sb8 1795	Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
|- F/ y ph   =>    |- ( A. x ph <-> A. y [ y / x ] ph )
Theorem	sb8e 1796	Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
|- F/ y ph   =>    |- ( E. x ph <-> E. y [ y / x ] ph )
1.4.4  Predicate calculus with distinct variables (cont.)
Theorem	ax16i 1797*	Inference with ax-16 1753 as its conclusion, that doesn't require ax-10 1451, ax-11 1452, or ax-12 1457 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.)
|- ( x = z -> ( ph <-> ps ) )   &    |- ( ps -> A. x ps )   =>    |- ( A. x x = y -> ( ph -> A. x ph ) )
Theorem	ax16ALT 1798*	Version of ax16 1752 that doesn't require ax-10 1451 or ax-12 1457 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A. x x = y -> ( ph -> A. x ph ) )
Theorem	spv 1799*	Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	spimev 1800*	Distinct-variable version of spime 1687. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph -> ps ) )   =>    |- ( ph -> E. x ps )