P. 17 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 1601-1700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nfa2 1601	Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x A. y A. x ph
Theorem	nfia1 1602	Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ( A. x ph -> A. x ps )
Theorem	19.21ht 1603	Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (New usage is discouraged.)
|- ( A. x ( ph -> A. x ph ) -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )
Theorem	19.21t 1604	Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.)
|- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )
Theorem	19.21 1605	Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in ph". (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   =>    |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )
Theorem	stdpc5 1606	An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis F/ x ph can be thought of as emulating " x is not free in ph". With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example x would not (for us) be free in x = x by nfequid 1724. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
|- F/ x ph   =>    |- ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) )
Theorem	nfimd 1607	If in a context x is not free in ps and ch, then it is not free in ( ps -> ch ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
|- ( ph -> F/ x ps )   &    |- ( ph -> F/ x ch )   =>    |- ( ph -> F/ x ( ps -> ch ) )
Theorem	aaanh 1608	Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
|- ( ph -> A. y ph )   &    |- ( ps -> A. x ps )   =>    |- ( A. x A. y ( ph /\ ps ) <-> ( A. x ph /\ A. y ps ) )
Theorem	aaan 1609	Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
|- F/ y ph   &    |- F/ x ps   =>    |- ( A. x A. y ( ph /\ ps ) <-> ( A. x ph /\ A. y ps ) )
Theorem	nfbid 1610	If in a context x is not free in ps and ch, then it is not free in ( ps <-> ch ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)
|- ( ph -> F/ x ps )   &    |- ( ph -> F/ x ch )   =>    |- ( ph -> F/ x ( ps <-> ch ) )
Theorem	nfbi 1611	If x is not free in ph and ps, then it is not free in ( ph <-> ps ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
|- F/ x ph   &    |- F/ x ps   =>    |- F/ x ( ph <-> ps )
1.3.7  The existential quantifier
Theorem	19.8a 1612	If a wff is true, then it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> E. x ph )
Theorem	19.8ad 1613	If a wff is true, it is true for at least one instance. Deduction form of 19.8a 1612. (Contributed by DAW, 13-Feb-2017.)
|- ( ph -> ps )   =>    |- ( ph -> E. x ps )
Theorem	19.23bi 1614	Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( E. x ph -> ps )   =>    |- ( ph -> ps )
Theorem	exlimih 1615	Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ps -> A. x ps )   &    |- ( ph -> ps )   =>    |- ( E. x ph -> ps )
Theorem	exlimi 1616	Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ps   &    |- ( ph -> ps )   =>    |- ( E. x ph -> ps )
Theorem	exlimd2 1617	Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1618 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x ps -> ch ) )
Theorem	exlimdh 1618	Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.)
|- ( ph -> A. x ph )   &    |- ( ch -> A. x ch )   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x ps -> ch ) )
Theorem	exlimd 1619	Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 18-Jun-2018.)
|- F/ x ph   &    |- F/ x ch   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x ps -> ch ) )
Theorem	exlimiv 1620*	Inference from Theorem 19.23 of [Margaris] p. 90. This inference, along with our many variants is used to implement a metatheorem called "Rule C" that is given in many logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or Rule C in Hirst and Hirst's A Primer for Logic and Proof p. 59 (PDF p. 65) at http://www.mathsci.appstate.edu/~jlh/primer/hirst.pdf. In informal proofs, the statement "Let C be an element such that..." almost always means an implicit application of Rule C. In essence, Rule C states that if we can prove that some element x exists satisfying a wff, i.e. E. x ph ( x ) where ph ( x ) has x free, then we can use ph ( C ) as a hypothesis for the proof where C is a new (ficticious) constant not appearing previously in the proof, nor in any axioms used, nor in the theorem to be proved. The purpose of Rule C is to get rid of the existential quantifier. We cannot do this in Metamath directly. Instead, we use the original ph (containing x) as an antecedent for the main part of the proof. We eventually arrive at ( ph -> ps ) where ps is the theorem to be proved and does not contain x. Then we apply exlimiv 1620 to arrive at ( E. x ph -> ps ). Finally, we separately prove E. x ph and detach it with modus ponens ax-mp 5 to arrive at the final theorem ps. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 25-Jul-2012.)
|- ( ph -> ps )   =>    |- ( E. x ph -> ps )
Theorem	exim 1621	Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
|- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )
Theorem	eximi 1622	Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ps )   =>    |- ( E. x ph -> E. x ps )
Theorem	2eximi 1623	Inference adding 2 existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
|- ( ph -> ps )   =>    |- ( E. x E. y ph -> E. x E. y ps )
Theorem	eximii 1624	Inference associated with eximi 1622. (Contributed by BJ, 3-Feb-2018.)
|- E. x ph   &    |- ( ph -> ps )   =>    |- E. x ps
Theorem	alinexa 1625	A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
|- ( A. x ( ph -> -. ps ) <-> -. E. x ( ph /\ ps ) )
Theorem	exbi 1626	Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ( ph <-> ps ) -> ( E. x ph <-> E. x ps ) )
Theorem	exbii 1627	Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
|- ( ph <-> ps )   =>    |- ( E. x ph <-> E. x ps )
Theorem	2exbii 1628	Inference adding 2 existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
|- ( ph <-> ps )   =>    |- ( E. x E. y ph <-> E. x E. y ps )
Theorem	3exbii 1629	Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
|- ( ph <-> ps )   =>    |- ( E. x E. y E. z ph <-> E. x E. y E. z ps )
Theorem	exancom 1630	Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
|- ( E. x ( ph /\ ps ) <-> E. x ( ps /\ ph ) )
Theorem	alrimdd 1631	Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> F/ x ps )   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ps -> A. x ch ) )
Theorem	alrimd 1632	Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- F/ x ps   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ps -> A. x ch ) )
Theorem	eximdh 1633	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x ps -> E. x ch ) )
Theorem	eximd 1634	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x ps -> E. x ch ) )
Theorem	nexd 1635	Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
|- ( ph -> A. x ph )   &    |- ( ph -> -. ps )   =>    |- ( ph -> -. E. x ps )
Theorem	exbidh 1636	Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x ps <-> E. x ch ) )
Theorem	albid 1637	Formula-building rule for universal quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps <-> A. x ch ) )
Theorem	exbid 1638	Formula-building rule for existential quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x ps <-> E. x ch ) )
Theorem	exsimpl 1639	Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( E. x ( ph /\ ps ) -> E. x ph )
Theorem	exsimpr 1640	Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( E. x ( ph /\ ps ) -> E. x ps )
Theorem	alexdc 1641	Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1667. (Contributed by Jim Kingdon, 2-Jun-2018.)
|- ( A. xDECID ph -> ( A. x ph <-> -. E. x -. ph ) )
Theorem	19.29 1642	Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( A. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) )
Theorem	19.29r 1643	Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
|- ( ( E. x ph /\ A. x ps ) -> E. x ( ph /\ ps ) )
Theorem	19.29r2 1644	Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.)
|- ( ( E. x E. y ph /\ A. x A. y ps ) -> E. x E. y ( ph /\ ps ) )
Theorem	19.29x 1645	Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
|- ( ( E. x A. y ph /\ A. x E. y ps ) -> E. x E. y ( ph /\ ps ) )
Theorem	19.35-1 1646	Forward direction of Theorem 19.35 of [Margaris] p. 90. The converse holds for classical logic but not (for all propositions) in intuitionistic logic. (Contributed by Mario Carneiro, 2-Feb-2015.)
|- ( E. x ( ph -> ps ) -> ( A. x ph -> E. x ps ) )
Theorem	19.35i 1647	Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
|- E. x ( ph -> ps )   =>    |- ( A. x ph -> E. x ps )
Theorem	19.25 1648	Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
|- ( A. y E. x ( ph -> ps ) -> ( E. y A. x ph -> E. y E. x ps ) )
Theorem	19.30dc 1649	Theorem 19.30 of [Margaris] p. 90, with an additional decidability condition. (Contributed by Jim Kingdon, 21-Jul-2018.)
|- (DECID E. x ps -> ( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) ) )
Theorem	19.43 1650	Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
|- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) )
Theorem	19.33b2 1651	The antecedent provides a condition implying the converse of 19.33 1506. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1652 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.)
|- ( ( -. E. x ph \/ -. E. x ps ) -> ( A. x ( ph \/ ps ) <-> ( A. x ph \/ A. x ps ) ) )
Theorem	19.33bdc 1652	Converse of 19.33 1506 given -. ( E. x ph /\ E. x ps ) and a decidability condition. Compare Theorem 19.33 of [Margaris] p. 90. For a version which does not require a decidability condition, see 19.33b2 1651 (Contributed by Jim Kingdon, 23-Apr-2018.)
|- (DECID E. x ph -> ( -. ( E. x ph /\ E. x ps ) -> ( A. x ( ph \/ ps ) <-> ( A. x ph \/ A. x ps ) ) ) )
Theorem	19.40 1653	Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) )
Theorem	19.40-2 1654	Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( E. x E. y ( ph /\ ps ) -> ( E. x E. y ph /\ E. x E. y ps ) )
Theorem	exintrbi 1655	Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
|- ( A. x ( ph -> ps ) -> ( E. x ph <-> E. x ( ph /\ ps ) ) )
Theorem	exintr 1656	Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
|- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ( ph /\ ps ) ) )
Theorem	alsyl 1657	Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)
|- ( ( A. x ( ph -> ps ) /\ A. x ( ps -> ch ) ) -> A. x ( ph -> ch ) )
Theorem	hbex 1658	If x is not free in ph, it is not free in E. y ph. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
|- ( ph -> A. x ph )   =>    |- ( E. y ph -> A. x E. y ph )
Theorem	nfex 1659	If x is not free in ph, it is not free in E. y ph. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
|- F/ x ph   =>    |- F/ x E. y ph
Theorem	19.2 1660	Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. (Contributed by O'Cat, 31-Mar-2008.)
|- ( A. x ph -> E. y ph )
Theorem	i19.24 1661	Theorem 19.24 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1646, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.)
|- ( ( A. x ph -> E. x ps ) -> E. x ( ph -> ps ) )   =>    |- ( ( A. x ph -> A. x ps ) -> E. x ( ph -> ps ) )
Theorem	i19.39 1662	Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1646, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.)
|- ( ( A. x ph -> E. x ps ) -> E. x ( ph -> ps ) )   =>    |- ( ( E. x ph -> E. x ps ) -> E. x ( ph -> ps ) )
Theorem	19.9ht 1663	A closed version of one direction of 19.9 1666. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> ph ) )
Theorem	19.9t 1664	A closed version of 19.9 1666. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.)
|- ( F/ x ph -> ( E. x ph <-> ph ) )
Theorem	19.9h 1665	A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.)
|- ( ph -> A. x ph )   =>    |- ( E. x ph <-> ph )
Theorem	19.9 1666	A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
|- F/ x ph   =>    |- ( E. x ph <-> ph )
Theorem	alexim 1667	One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1641. (Contributed by Jim Kingdon, 2-Jul-2018.)
|- ( A. x ph -> -. E. x -. ph )
Theorem	exnalim 1668	One direction of Theorem 19.14 of [Margaris] p. 90. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)
|- ( E. x -. ph -> -. A. x ph )
Theorem	exanaliim 1669	A transformation of quantifiers and logical connectives. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)
|- ( E. x ( ph /\ -. ps ) -> -. A. x ( ph -> ps ) )
Theorem	alexnim 1670	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 1671	The double negation of a universal quantification implies the universal quantification of the double negation. (Contributed by BJ, 24-Nov-2023.)
|- ( -. -. A. x ph -> A. x -. -. ph )
Theorem	ax6blem 1672	If x is not free in ph, it is not free in -. ph. This theorem doesn't use ax6b 1673 compared to hbnt 1675. (Contributed by GD, 27-Jan-2018.)
|- ( ph -> A. x ph )   =>    |- ( -. ph -> A. x -. ph )
Theorem	ax6b 1673	Quantified Negation. Axiom C5-2 of [Monk2] p. 113. (Contributed by GD, 27-Jan-2018.)
|- ( -. A. x ph -> A. x -. A. x ph )
Theorem	hbn1 1674	x is not free in -. A. x ph. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
|- ( -. A. x ph -> A. x -. A. x ph )
Theorem	hbnt 1675	Closed theorem version of bound-variable hypothesis builder hbn 1676. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
|- ( A. x ( ph -> A. x ph ) -> ( -. ph -> A. x -. ph ) )
Theorem	hbn 1676	If x is not free in ph, it is not free in -. ph. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   =>    |- ( -. ph -> A. x -. ph )
Theorem	hbnd 1677	Deduction form of bound-variable hypothesis builder hbn 1676. (Contributed by NM, 3-Jan-2002.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> ( -. ps -> A. x -. ps ) )
Theorem	nfnt 1678	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 1679	Deduction associated with nfnt 1678. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x -. ps )
Theorem	nfn 1680	Inference associated with nfnt 1678. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph   =>    |- F/ x -. ph
Theorem	nfdc 1681	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 1682	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 1683	A deduction version of one direction of 19.9 1666. (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 1684	A deduction version of one direction of 19.9 1666. This is an older variation of this theorem; new proofs should use 19.9d 1683. (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 1685	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 1686	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 1687	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 1688	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 1689	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 1690	An alternate definition of df-nf 1483, 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 1691	An alternate definition of df-nf 1483. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- ( F/ x ph <-> A. x ( E. x ph -> ph ) )
Theorem	nf4dc 1692	Variable x is effectively not free in ph iff ph is always true or always false, given a decidability condition. The reverse direction, nf4r 1693, 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 1693	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 1692. (Contributed by Jim Kingdon, 21-Jul-2018.)
|- ( ( A. x ph \/ A. x -. ph ) -> F/ x ph )
Theorem	19.36i 1694	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 1695	Closed form of 19.36i 1694. 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 1696	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 1697*	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 1698	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 1699	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 1700	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 ) )