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	alexnim 1701	A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.)
|- ( A. x E. y -. ph -> -. E. x A. y ph )
Theorem	nnal 1702	The double negation of a universal quantification implies the universal quantification of the double negation. The converse holds in classical but not in intuitionistic logic. (Contributed by BJ, 24-Nov-2023.)
|- ( -. -. A. x ph -> A. x -. -. ph )
Theorem	hbn 1703	If x is not free in ph, then it is not free in -. ph. This theorem does not depend on ax-ial 1587, contrary to hbn1 1704 and hbnt 1705. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-ial 1587. (Revised by GD, 27-Jan-2018.)
|- ( ph -> A. x ph )   =>    |- ( -. ph -> A. x -. ph )
Theorem	hbn1 1704	Quantified Negation. Axiom C5-2 of [Monk2] p. 113. The setvar x is not free in -. A. x ph. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.) (Proof shortened by GD, 27-Jan-2018.)
|- ( -. A. x ph -> A. x -. A. x ph )
Theorem	hbnt 1705	Closed form of bound-variable hypothesis builder hbn 1703. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
|- ( A. x ( ph -> A. x ph ) -> ( -. ph -> A. x -. ph ) )
Theorem	hbnOLD 1706	Obsolete proof of hbn 1703 as of 2-May-2026. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> A. x ph )   =>    |- ( -. ph -> A. x -. ph )
Theorem	hbnd 1707	Deduction form of bound-variable hypothesis builder hbn 1703. (Contributed by NM, 3-Jan-2002.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> ( -. ps -> A. x -. ps ) )
Theorem	nfnt 1708	If x is not free in ph, then it is not free in -. ph. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.)
|- ( F/ x ph -> F/ x -. ph )
Theorem	nfnd 1709	Deduction associated with nfnt 1708. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x -. ps )
Theorem	nfn 1710	Inference associated with nfnt 1708. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph   =>    |- F/ x -. ph
Theorem	nfdc 1711	If x is not free in ph, it is not free in DECID ph. (Contributed by Jim Kingdon, 11-Mar-2018.)
|- F/ x ph   =>    |- F/ xDECID ph
Theorem	modal-5 1712	The analog in our predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.)
|- ( -. A. x -. ph -> A. x -. A. x -. ph )
Theorem	19.9d 1713	A deduction version of one direction of 19.9 1697. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
|- ( ps -> F/ x ph )   =>    |- ( ps -> ( E. x ph -> ph ) )
Theorem	19.9hd 1714	A deduction version of one direction of 19.9 1697. This is an older variation of this theorem; new proofs should use 19.9d 1713. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
|- ( ps -> A. x ps )   &    |- ( ps -> ( ph -> A. x ph ) )   =>    |- ( ps -> ( E. x ph -> ph ) )
Theorem	excomim 1715	One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
|- ( E. x E. y ph -> E. y E. x ph )
Theorem	excom 1716	Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
|- ( E. x E. y ph <-> E. y E. x ph )
Theorem	19.12 1717	Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.)
|- ( E. x A. y ph -> A. y E. x ph )
Theorem	19.19 1718	Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- F/ x ph   =>    |- ( A. x ( ph <-> ps ) -> ( ph <-> E. x ps ) )
Theorem	19.21-2 1719	Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)
|- F/ x ph   &    |- F/ y ph   =>    |- ( A. x A. y ( ph -> ps ) <-> ( ph -> A. x A. y ps ) )
Theorem	nf2 1720	An alternate definition of df-nf 1514, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- ( F/ x ph <-> ( E. x ph -> A. x ph ) )
Theorem	nf3 1721	An alternate definition of df-nf 1514. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- ( F/ x ph <-> A. x ( E. x ph -> ph ) )
Theorem	nf4dc 1722	Variable x is effectively not free in ph iff ph is always true or always false, given a decidability condition. The reverse direction, nf4r 1723, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
|- (DECID E. x ph -> ( F/ x ph <-> ( A. x ph \/ A. x -. ph ) ) )
Theorem	nf4r 1723	If ph is always true or always false, then variable x is effectively not free in ph. The converse holds given a decidability condition, as seen at nf4dc 1722. (Contributed by Jim Kingdon, 21-Jul-2018.)
|- ( ( A. x ph \/ A. x -. ph ) -> F/ x ph )
Theorem	19.36i 1724	Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
|- F/ x ps   &    |- E. x ( ph -> ps )   =>    |- ( A. x ph -> ps )
Theorem	19.36-1 1725	Closed form of 19.36i 1724. One direction of Theorem 19.36 of [Margaris] p. 90. The converse holds in classical logic, but does not hold (for all propositions) in intuitionistic logic. (Contributed by Jim Kingdon, 20-Jun-2018.)
|- F/ x ps   =>    |- ( E. x ( ph -> ps ) -> ( A. x ph -> ps ) )
Theorem	19.37-1 1726	One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.)
|- F/ x ph   =>    |- ( E. x ( ph -> ps ) -> ( ph -> E. x ps ) )
Theorem	19.37aiv 1727*	Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- E. x ( ph -> ps )   =>    |- ( ph -> E. x ps )
Theorem	19.38 1728	Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
Theorem	19.23t 1729	Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
|- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )
Theorem	19.23 1730	Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
|- F/ x ps   =>    |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) )
Theorem	19.32dc 1731	Theorem 19.32 of [Margaris] p. 90, where ph is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
|- F/ x ph   =>    |- (DECID ph -> ( A. x ( ph \/ ps ) <-> ( ph \/ A. x ps ) ) )
Theorem	19.32r 1732	One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if ph is decidable, as seen at 19.32dc 1731. (Contributed by Jim Kingdon, 28-Jul-2018.)
|- F/ x ph   =>    |- ( ( ph \/ A. x ps ) -> A. x ( ph \/ ps ) )
Theorem	19.31r 1733	One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
|- F/ x ps   =>    |- ( ( A. x ph \/ ps ) -> A. x ( ph \/ ps ) )
Theorem	19.44 1734	Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- F/ x ps   =>    |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ ps ) )
Theorem	19.45 1735	Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- F/ x ph   =>    |- ( E. x ( ph \/ ps ) <-> ( ph \/ E. x ps ) )
Theorem	19.34 1736	Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ( A. x ph \/ E. x ps ) -> E. x ( ph \/ ps ) )
Theorem	19.41h 1737	Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1738 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
|- ( ps -> A. x ps )   =>    |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) )
Theorem	19.41 1738	Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
|- F/ x ps   =>    |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) )
Theorem	19.42h 1739	Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1740 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)
|- ( ph -> A. x ph )   =>    |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )
Theorem	19.42 1740	Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
|- F/ x ph   =>    |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )
Theorem	excom13 1741	Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
|- ( E. x E. y E. z ph <-> E. z E. y E. x ph )
Theorem	exrot3 1742	Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
|- ( E. x E. y E. z ph <-> E. y E. z E. x ph )
Theorem	exrot4 1743	Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
|- ( E. x E. y E. z E. w ph <-> E. z E. w E. x E. y ph )
Theorem	nexr 1744	Inference from 19.8a 1643. (Contributed by Jeff Hankins, 26-Jul-2009.)
|- -. E. x ph   =>    |- -. ph
Theorem	exan 1745	Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( E. x ph /\ ps )   =>    |- E. x ( ph /\ ps )
Theorem	hbexd 1746	Deduction form of bound-variable hypothesis builder hbex 1689. (Contributed by NM, 2-Jan-2002.)
|- ( ph -> A. y ph )   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> ( E. y ps -> A. x E. y ps ) )
Theorem	eeor 1747	Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
|- F/ y ph   &    |- F/ x ps   =>    |- ( E. x E. y ( ph \/ ps ) <-> ( E. x ph \/ E. y ps ) )
1.3.8  Equality theorems without distinct variables
Theorem	a9e 1748	At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1500 through ax-14 2212 and ax-17 1579, all axioms other than ax-9 1584 are believed to be theorems of free logic, although the system without ax-9 1584 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
|- E. x x = y
Theorem	a9ev 1749*	At least one individual exists. Weaker version of a9e 1748. (Contributed by NM, 3-Aug-2017.)
|- E. x x = y
Theorem	ax9o 1750	An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
|- ( A. x ( x = y -> A. x ph ) -> ph )
Theorem	spimfv 1751*	Specialization, using implicit substitution. Version of spim 1791 with a disjoint variable condition. See spimv 1864 for another variant. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 31-May-2019.)
|- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	chvarfv 1752*	Implicit substitution of y for x into a theorem. Version of chvar 1810 with a disjoint variable condition. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by BJ, 31-May-2019.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	equid 1753	Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms. This proof is similar to Tarski's and makes use of a dummy variable y. It also works in intuitionistic logic, unlike some other possible ways of proving this theorem. (Contributed by NM, 1-Apr-2005.)
|- x = x
Theorem	nfequid 1754	Bound-variable hypothesis builder for x = x. This theorem tells us that any variable, including x, is effectively not free in x = x, even though x is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)
|- F/ y x = x
Theorem	stdpc6 1755	One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1823.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)
|- A. x x = x
Theorem	equcomi 1756	Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> y = x )
Theorem	ax6evr 1757*	A commuted form of a9ev 1749. The naming reflects how axioms were numbered in the Metamath Proof Explorer as of 2020 (a numbering which we eventually plan to adopt here too, but until this happens everywhere only some theorems will have it). (Contributed by BJ, 7-Dec-2020.)
|- E. x y = x
Theorem	equcom 1758	Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
|- ( x = y <-> y = x )
Theorem	equcomd 1759	Deduction form of equcom 1758, symmetry of equality. For the versions for classes, see eqcom 2240 and eqcomd 2244. (Contributed by BJ, 6-Oct-2019.)
|- ( ph -> x = y )   =>    |- ( ph -> y = x )
Theorem	equcoms 1760	An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ph )   =>    |- ( y = x -> ph )
Theorem	equtr 1761	A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
|- ( x = y -> ( y = z -> x = z ) )
Theorem	equtrr 1762	A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)
|- ( x = y -> ( z = x -> z = y ) )
Theorem	equtr2 1763	A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ( x = z /\ y = z ) -> x = y )
Theorem	equequ1 1764	An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( x = z <-> y = z ) )
Theorem	equequ2 1765	An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( z = x <-> z = y ) )
Theorem	ax11i 1766	Inference that has ax-11 1559 (without A. y) as its conclusion and does not require ax-10 1558, ax-11 1559, or ax12 1565 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( ps -> A. x ps )   =>    |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
1.3.9  Axioms ax-10 and ax-11
Theorem	ax10o 1767	Show that ax-10o 1768 can be derived from ax-10 1558. An open problem is whether this theorem can be derived from ax-10 1558 and the others when ax-11 1559 is replaced with ax-11o 1876. See Theorem ax10 1769 for the rederivation of ax-10 1558 from ax10o 1767. Normally, ax10o 1767 should be used rather than ax-10o 1768, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)
|- ( A. x x = y -> ( A. x ph -> A. y ph ) )
Axiom	ax-10o 1768	Axiom ax-10o 1768 ("o" for "old") was the original version of ax-10 1558, before it was discovered (in May 2008) that the shorter ax-10 1558 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint). This axiom is redundant, as shown by Theorem ax10o 1767. Normally, ax10o 1767 should be used rather than ax-10o 1768, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
|- ( A. x x = y -> ( A. x ph -> A. y ph ) )
Theorem	ax10 1769	Rederivation of ax-10 1558 from original version ax-10o 1768. See Theorem ax10o 1767 for the derivation of ax-10o 1768 from ax-10 1558. This theorem should not be referenced in any proof. Instead, use ax-10 1558 above so that uses of ax-10 1558 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)
|- ( A. x x = y -> A. y y = x )
Theorem	hbae 1770	All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
|- ( A. x x = y -> A. z A. x x = y )
Theorem	nfae 1771	All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ z A. x x = y
Theorem	hbaes 1772	Rule that applies hbae 1770 to antecedent. (Contributed by NM, 5-Aug-1993.)
|- ( A. z A. x x = y -> ph )   =>    |- ( A. x x = y -> ph )
Theorem	hbnae 1773	All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.)
|- ( -. A. x x = y -> A. z -. A. x x = y )
Theorem	nfnae 1774	All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ z -. A. x x = y
Theorem	hbnaes 1775	Rule that applies hbnae 1773 to antecedent. (Contributed by NM, 5-Aug-1993.)
|- ( A. z -. A. x x = y -> ph )   =>    |- ( -. A. x x = y -> ph )
Theorem	naecoms 1776	A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
|- ( -. A. x x = y -> ph )   =>    |- ( -. A. y y = x -> ph )
Theorem	equs4 1777	Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)
|- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
Theorem	equsalh 1778	A useful equivalence related to substitution. New proofs should use equsal 1779 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ( x = y -> ph ) <-> ps )
Theorem	equsal 1779	A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ( x = y -> ph ) <-> ps )
Theorem	equsex 1780	A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ( x = y /\ ph ) <-> ps )
Theorem	equsexvw 1781*	Version of equsex 1780 with two disjoint variable conditions. (Contributed by BJ, 31-May-2019.) (Proof shortened by Wolf Lammen, 23-Oct-2023.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ( x = y /\ ph ) <-> ps )
Theorem	equsexd 1782	Deduction form of equsex 1780. (Contributed by Jim Kingdon, 29-Dec-2017.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( E. x ( x = y /\ ps ) <-> ch ) )
Theorem	dral1 1783	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, 24-Nov-1994.)
|- ( A. x x = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( A. x ph <-> A. y ps ) )
Theorem	dral2 1784	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 -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( A. z ph <-> A. z ps ) )
Theorem	drex2 1785	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 -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( E. z ph <-> E. z ps ) )
Theorem	drnf1 1786	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
|- ( A. x x = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) )
Theorem	drnf2 1787	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
|- ( A. x x = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y -> ( F/ z ph <-> F/ z ps ) )
Theorem	spimth 1788	Closed theorem form of spim 1791. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.)
|- ( A. x ( ( ps -> A. x ps ) /\ ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) )
Theorem	spimt 1789	Closed theorem form of spim 1791. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.)
|- ( ( F/ x ps /\ A. x ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) )
Theorem	spimh 1790	Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1791 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 8-May-2008.) (New usage is discouraged.)
|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	spim 1791	Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1791 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
|- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	spimeh 1792	Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.)
|- ( ph -> A. x ph )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( ph -> E. x ps )
Theorem	spimed 1793	Deduction version of spime 1794. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)
|- ( ch -> F/ x ph )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( ch -> ( ph -> E. x ps ) )
Theorem	spime 1794	Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.)
|- F/ x ph   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( ph -> E. x ps )
Theorem	cbv3 1795	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because proofs are encouraged to use the weaker cbv3v 1797 if possible. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> A. y ps )
Theorem	cbv3h 1796	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.)
|- ( ph -> A. y ph )   &    |- ( ps -> A. x ps )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> A. y ps )
Theorem	cbv3v 1797*	Rule used to change bound variables, using implicit substitution. Version of cbv3 1795 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> A. y ps )
Theorem	cbv1 1798	Rule used to change bound variables, using implicit substitution. Revised to format hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
|- F/ x ph   &    |- F/ y ph   &    |- ( ph -> F/ y ps )   &    |- ( ph -> F/ x ch )   &    |- ( ph -> ( x = y -> ( ps -> ch ) ) )   =>    |- ( ph -> ( A. x ps -> A. y ch ) )
Theorem	cbv1h 1799	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)
|- ( ph -> ( ps -> A. y ps ) )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( x = y -> ( ps -> ch ) ) )   =>    |- ( A. x A. y ph -> ( A. x ps -> A. y ch ) )
Theorem	cbv1v 1800*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 16-Jun-2019.)
|- F/ x ph   &    |- F/ y ph   &    |- ( ph -> F/ y ps )   &    |- ( ph -> F/ x ch )   &    |- ( ph -> ( x = y -> ( ps -> ch ) ) )   =>    |- ( ph -> ( A. x ps -> A. y ch ) )