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	sbieh 1801	Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1802 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 1802	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 1803*	Conversion of implicit substitution to explicit substitution. Version of sbie 1802 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 1804*	An equivalence related to implicit substitution. Version of equsal 1738 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 1805	A property related to substitution that unlike equs5 1840 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 1806	A property related to substitution that unlike equs5 1840 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 1807	Analogue to ax-11 1517 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 1808	Quantifier Substitution for existential quantifiers. Analogue to ax10o 1726 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 1809	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 1810	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 1811	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 1812	A version of sb4 1843 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 1813	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 1814	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 1815	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 1816	One direction of a simplified definition of substitution that unlike sb4 1843 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
|- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) )
Theorem	hbsb2a 1817	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 1818	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 1819	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 1820	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 1821	Version of sbco 1980 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 1822*	A version of spim 1749 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 1823*	A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1825. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
|- ( A. x x = y -> A. z w = v )
Theorem	ax16 1824*	Theorem showing that ax-16 1825 is redundant if ax-17 1537 is included in the axiom system. The important part of the proof is provided by aev 1823. See ax16ALT 1870 for an alternate proof that does not require ax-10 1516 or ax12 1523. This theorem should not be referenced in any proof. Instead, use ax-16 1825 below so that theorems needing ax-16 1825 can be more easily identified. (Contributed by NM, 8-Nov-2006.)
|- ( A. x x = y -> ( ph -> A. x ph ) )
Axiom	ax-16 1825*	Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1537 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 1537; see Theorem ax16 1824. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax16 1824. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
|- ( A. x x = y -> ( ph -> A. x ph ) )
Theorem	dveeq2 1826*	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 1827*	Quantifier introduction when one pair of variables is distinct. Like dveeq2 1826 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 1828*	Proof of dvelimf 2027 using dveeq2 1826 (shown as the last hypothesis) instead of ax12 1523. This shows that ax12 1523 could be replaced by dveeq2 1826 (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 1829*	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 1830*	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 1831*	Recovery of ax11o 1833 from ax11v 1838 without using ax-11 1517. The hypothesis is even weaker than ax11v 1838, 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 1833. (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 1832*	Derive ax-11o 1834 from a hypothesis in the form of ax-11 1517. The hypothesis is even weaker than ax-11 1517, 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 1833. (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 1833	Derivation of set.mm's original ax-11o 1834 from the shorter ax-11 1517 that has replaced it. An open problem is whether this theorem can be proved without relying on ax-16 1825 or ax-17 1537. Normally, ax11o 1833 should be used rather than ax-11o 1834, 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 1834	Axiom ax-11o 1834 ("o" for "old") was the original version of ax-11 1517, before it was discovered (in Jan. 2007) that the shorter ax-11 1517 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 1833. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax11o 1833. (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 1835*	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 1836*	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 1837	A bidirectional version of ax-11o 1834. (Contributed by NM, 30-Jun-2006.)
|- ( ( -. A. x x = y /\ x = y ) -> ( ph <-> A. x ( x = y -> ph ) ) )
Theorem	ax11v 1838*	This is a version of ax-11o 1834 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 1839*	Analogue to ax11v 1838 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
|- ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) )
Theorem	equs5 1840	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 1841	Lemma used in proofs of substitution properties. Like equs5 1840 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 1842	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 1843	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 1844	One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1843 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 1845	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 1846	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 1847	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 1848	Bound-variable hypothesis builder for substitution. Similar to hbsb2 1847 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 1849	Propositional logic lemma used in the sbequi 1850 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
|- ( ph \/ ( ps -> ( ch -> th ) ) )   &    |- ( ta \/ ( ps -> ( th -> et ) ) )   =>    |- ( ph \/ ( ta \/ ( ps -> ( ch -> et ) ) ) )
Theorem	sbequi 1850	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 1851	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 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, 27-Feb-2005.)
|- ( A. x x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) )
Theorem	spsbe 1853	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 1854	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 1855	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 1856	Deduction substituting both sides of a biconditional. New proofs should use sbbid 1857 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 1857	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 1858	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 1859	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 1860	An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   =>    |- ( [ y / x ] [ x / y ] ph <-> ph )
Theorem	sbid2 1861	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 1862	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 1863	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 1864	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 1865	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 1866	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 1867	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 1868	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 1869*	Inference with ax-16 1825 as its conclusion, that does not require ax-10 1516, ax-11 1517, or ax12 1523 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 1870*	Version of ax16 1824 that does not require ax-10 1516 or ax12 1523 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 1871*	Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	spimev 1872*	Distinct-variable version of spime 1752. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( ph -> ps ) )   =>    |- ( ph -> E. x ps )
Theorem	speiv 1873*	Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ps   =>    |- E. x ph
Theorem	equvin 1874*	A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)
|- ( x = y <-> E. z ( x = z /\ z = y ) )
Theorem	a16g 1875*	A generalization of Axiom ax-16 1825. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( A. x x = y -> ( ph -> A. z ph ) )
Theorem	a16gb 1876*	A generalization of Axiom ax-16 1825. (Contributed by NM, 5-Aug-1993.)
|- ( A. x x = y -> ( ph <-> A. z ph ) )
Theorem	a16nf 1877*	If there is only one element in the universe, then everything satisfies F/. (Contributed by Mario Carneiro, 7-Oct-2016.)
|- ( A. x x = y -> F/ z ph )
Theorem	2albidv 1878*	Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x A. y ps <-> A. x A. y ch ) )
Theorem	2exbidv 1879*	Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x E. y ps <-> E. x E. y ch ) )
Theorem	3exbidv 1880*	Formula-building rule for 3 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x E. y E. z ps <-> E. x E. y E. z ch ) )
Theorem	4exbidv 1881*	Formula-building rule for 4 existential quantifiers (deduction form). (Contributed by NM, 3-Aug-1995.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x E. y E. z E. w ps <-> E. x E. y E. z E. w ch ) )
Theorem	19.9v 1882*	Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 21-May-2007.)
|- ( E. x ph <-> ph )
Theorem	exlimdd 1883	Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- F/ x ph   &    |- F/ x ch   &    |- ( ph -> E. x ps )   &    |- ( ( ph /\ ps ) -> ch )   =>    |- ( ph -> ch )
Theorem	19.21v 1884*	Special case of Theorem 19.21 of [Margaris] p. 90. Notational convention: We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as ( ph -> A. x ph ) in 19.21 1594 via the use of distinct variable conditions combined with ax-17 1537. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g., euf 2043 derived from df-eu 2041. The "f" stands for "not free in" which is less restrictive than "does not occur in". (Contributed by NM, 5-Aug-1993.)
|- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )
Theorem	alrimiv 1885*	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ps )   =>    |- ( ph -> A. x ps )
Theorem	alrimivv 1886*	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
|- ( ph -> ps )   =>    |- ( ph -> A. x A. y ps )
Theorem	alrimdv 1887*	Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ps -> A. x ch ) )
Theorem	nfdv 1888*	Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> F/ x ps )
Theorem	2ax17 1889*	Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
|- ( ph -> A. x A. y ph )
Theorem	alimdv 1890*	Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> A. x ch ) )
Theorem	eximdv 1891*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x ps -> E. x ch ) )
Theorem	2alimdv 1892*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x A. y ps -> A. x A. y ch ) )
Theorem	2eximdv 1893*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x E. y ps -> E. x E. y ch ) )
Theorem	19.23v 1894*	Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.)
|- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) )
Theorem	19.23vv 1895*	Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.)
|- ( A. x A. y ( ph -> ps ) <-> ( E. x E. y ph -> ps ) )
Theorem	sbbidv 1896*	Deduction substituting both sides of a biconditional, with ph and x disjoint. See also sbbid 1857. (Contributed by Wolf Lammen, 6-May-2023.) (Proof shortened by Steven Nguyen, 6-Jul-2023.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( [ t / x ] ps <-> [ t / x ] ch ) )
Theorem	sb56 1897*	Two equivalent ways of expressing the proper substitution of y for x in ph, when x and y are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1774. (Contributed by NM, 14-Apr-2008.)
|- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
Theorem	sb6 1898*	Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)
|- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
Theorem	sb5 1899*	Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)
|- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
Theorem	sbnv 1900*	Version of sbn 1964 where x and y are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)
|- ( [ y / x ] -. ph <-> -. [ y / x ] ph )