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Theorem List for Metamath Proof Explorer - 1701-1800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremax12dgen3 1701 Degenerate instance of ax-12 1866 where bundled variables  y and  z have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  y  ->  A. x  y  =  y )
 )
 
Theoremax12dgen4 1702 Degenerate instance of ax-12 1866 where bundled variables  x,  y, and  z have a common substitution. Uses only Tarski's FOL axiom schemes . (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  x  ->  ( x  =  x  ->  A. x  x  =  x )
 )
 
1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)

In this section we introduce four additional schemes ax-6 1703, ax-7 1708, ax-11 1715, and ax-12 1866 that are not part of Tarski's system but can be proved (outside of Metamath) as theorem schemes of Tarski's system. These are needed to give our system the property of "metalogical completeness," which means that we can prove (with Metamath) all possible schemes expressible in our language of wff metavariables ranging over object-language wffs and set metavariables ranging over object-language individual variables.

To show that these schemes are valid metatheorems of Tarski's system S2, above we proved from Tarski's system theorems ax6w 1691, ax7w 1692, ax12w 1698, and ax11w 1695, which show that any object-language instance of these schemes (emulated by having no wff metamavariables and requiring all set metamavariables to be mutually distinct) can be proved using only the schemes in Tarski's system S2.

An open problem is to show that these four additional schemes are metalogically independent from Tarski's. So far, independence of ax-11 1715 from all others has been shown, and independence of Tarski's ax-9 1635 from all others has been shown.

 
1.5.1  Axiom scheme ax-6 (Quantified Negation)
 
Axiomax-6 1703 Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax6w 1691) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremhbn1 1704  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 )
 
Theoremhbe1 1705  x is not free in  E. x ph. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x ph  ->  A. x E. x ph )
 
Theoremnfe1 1706  x is not free in  E. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x E. x ph
 
Theoremmodal-5 1707 The analog in our "pure" predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.)
 |-  ( -.  A. x  -.  ph  ->  A. x  -.  A. x  -.  ph )
 
1.5.2  Axiom scheme ax-7 (Quantifier Commutation)
 
Axiomax-7 1708 Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. One of the 4 axioms of pure predicate calculus. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax7w 1692) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theorema7s 1709 Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  ->  ps )   =>    |-  ( A. y A. x ph  ->  ps )
 
Theoremhbal 1710 If  x is not free in  ph, it is not free in  A. y ph. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. y ph  ->  A. x A. y ph )
 
Theoremalcom 1711 Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  <->  A. y A. x ph )
 
Theoremalrot3 1712 Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y A. z ph  <->  A. y A. z A. x ph )
 
Theoremalrot4 1713 Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
 |-  ( A. x A. y A. z A. w ph  <->  A. z A. w A. x A. y ph )
 
Theoremhbald 1714 Deduction form of bound-variable hypothesis builder hbal 1710. (Contributed by NM, 2-Jan-2002.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  ( A. y ps  ->  A. x A. y ps ) )
 
1.5.3  Axiom scheme ax-11 (Substitution)
 
Axiomax-11 1715 Axiom of Substitution. One of the 5 equality axioms of predicate calculus. The final consequent  A. x ( x  =  y  ->  ph ) is a way of expressing " y substituted for  x in wff  ph " (cf. sb6 2038). 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.

The original version of this axiom was ax-11o 2080 ("o" for "old") and was replaced with this shorter ax-11 1715 in Jan. 2007. The old axiom is proved from this one as theorem ax11o 1934. Conversely, this axiom is proved from ax-11o 2080 as theorem ax11 2094.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-11o 2080) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax11v 2036 and ax11v2 1932 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax11w 1695) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 22-Jan-2007.)

 |-  ( x  =  y 
 ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremsp 1716 Specialization. A universally quantified wff implies the wff without a quantifier Axiom scheme B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77). Also appears as Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint).

For the axiom of specialization presented in many logic textbooks, see theorem stdpc4 1964.

This theorem shows that our obsolete axiom ax-4 2074 can be derived from the others. The proof uses ideas from the proof of Lemma 21 of [Monk2] p. 114.

It appears that this scheme cannot be derived directly from Tarski's axioms without auxilliary axiom scheme ax-11 1715. It is thought the best we can do using only Tarski's axioms is spw 1660. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.)

 |-  ( A. x ph  -> 
 ph )
 
Theoremax5o 1717 Show that the original axiom ax-5o 2075 can be derived from ax-5 1544 and others. See ax5 2085 for the rederivation of ax-5 1544 from ax-5o 2075.

Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)

 |-  ( A. x (
 A. x ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theorem19.8a 1718 If a wff is true, 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 )
 
Theoremhba1 1719  x is not free in  A. x ph. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  ->  A. x A. x ph )
 
Theoremhbn 1720 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 )
 
Theoremhbimd 1721 Deduction form of bound-variable hypothesis builder hbim 1725. (Contributed by NM, 1-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   &    |-  ( ph  ->  ( ch  ->  A. x ch ) )   =>    |-  ( ph  ->  (
 ( ps  ->  ch )  ->  A. x ( ps 
 ->  ch ) ) )
 
Theoremspimeh 1722* Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.)
 |-  ( ph  ->  A. x ph )   &    |-  ( x  =  z  ->  ( ph  ->  ps ) )   =>    |-  ( ph  ->  E. x ps )
 
Theoremax6o 1723 Show that the original axiom ax-6o 2076 can be derived from ax-6 1703 and others. See ax6 2086 for the rederivation of ax-6 1703 from ax-6o 2076.

Normally, ax6o 1723 should be used rather than ax-6o 2076, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)

 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremhbnt 1724 Closed theorem version of bound-variable hypothesis builder hbn 1720. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph  ->  A. x ph )  ->  ( -.  ph  ->  A. x  -.  ph )
 )
 
Theoremhbim 1725 If  x is not free in  ph and  ps, it is not free in  ( ph  ->  ps ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ( ph  ->  ps )  ->  A. x ( ph  ->  ps )
 )
 
Theorem19.9ht 1726 A closed version of 19.9 1783. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph  ->  A. x ph )  ->  ( E. x ph  -> 
 ph ) )
 
Theorem19.9h 1727 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 )
 
Theorem19.23h 1728 Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |-  ( ps  ->  A. x ps )   =>    |-  ( A. x (
 ph  ->  ps )  <->  ( E. x ph 
 ->  ps ) )
 
Theoremexlimih 1729 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 )
 
Theoremequsalhw 1730* Weaker version of equsalh 1901 (requiring distinct variables) without using ax-12 1866. (Contributed by NM, 29-Nov-2015.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
 
Theorem19.21h 1731 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in  ph." (Contributed by NM, 1-Aug-2017.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. x (
 ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
 
Theoremhbim1 1732 A closed form of hbim 1725. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ( ph  ->  ps )  ->  A. x (
 ph  ->  ps ) )
 
Theoremhbex 1733 If  x is not free in  ph, it is not free in  E. y ph. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( E. y ph  ->  A. x E. y ph )
 
Theorem19.12 1734 Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! But sometimes the converse does hold, as in 19.12vv 1839 and r19.12sn 3696. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x A. y ph  ->  A. y E. x ph )
 
Theoremdvelimhw 1735* Proof of dvelimh 1904 without using ax-12 1866 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   &    |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremhban 1736 If  x is not free in  ph and  ps, it is not free in  ( ph  /\  ps ). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ( ph  /\ 
 ps )  ->  A. x ( ph  /\  ps )
 )
 
Theoremcbv3hv 1737* Lemma for ax10 1884. Similar to cbv3h 1923. Requires distinct variables but avoids ax-12 1866. (Contributed by NM, 25-Jul-2015.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremspi 1738 Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
 |- 
 A. x ph   =>    |-  ph
 
Theoremsps 1739 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theoremspsd 1740 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  ch ) )
 
Theoremnfr 1741 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
 |-  ( F/ x ph  ->  ( ph  ->  A. x ph ) )
 
Theoremnfri 1742 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |-  ( ph  ->  A. x ph )
 
Theoremnfrd 1743 Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  ( ps  ->  A. x ps )
 )
 
Theoremalimd 1744 Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( A. x ps  ->  A. x ch ) )
 
Theoremalrimi 1745 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x ps )
 
Theoremnfd 1746 Deduce that  x is not free in  ph in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  F/ x ps )
 
Theoremnfdh 1747 Deduce that  x is not free in  ph in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  F/ x ps )
 
Theoremalrimdd 1748 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 )
 )
 
Theoremalrimd 1749 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 )
 )
 
Theoremeximd 1750 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 ) )
 
Theoremnexd 1751 Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  -. 
 ps )   =>    |-  ( ph  ->  -.  E. x ps )
 
Theoremalbid 1752 Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theoremexbid 1753 Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. x ch )
 )
 
Theoremnfbidf 1754 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( F/ x ps  <->  F/ x ch )
 )
 
Theorema6e 1755 Abbreviated version of ax6o 1723. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x A. x ph  ->  ph )
 
Theoremnfa1 1756  x is not free in  A. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x A. x ph
 
Theoremnfnf1 1757  x is not free in  F/ x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x F/ x ph
 
Theorema5i 1758 Inference version of ax5o 1717. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
Theoremhb3an 1759 If  x is not free in  ph,  ps, and  ch, it is not free in  ( ph  /\  ps  /\  ch ). (Contributed by NM, 14-Sep-2003.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( ch  ->  A. x ch )   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  A. x (
 ph  /\  ps  /\  ch ) )
 
Theoremnfnd 1760 If  x is not free in  ph, it is not free in  -.  ph. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x  -.  ps )
 
Theoremnfimd 1761 If  x is not free in  ph and  ps, it is not free in  ( ph  ->  ps ). (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  F/ x ( ps  ->  ch ) )
 
Theoremnfbid 1762 If  x is not free in  ph and  ps, it is not free in  ( ph  ->  ps ). (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  F/ x ( ps  <->  ch ) )
 
Theoremnfand 1763 If  x is not free in  ph and  ps, it is not free in  ( ph  /\  ps ). (Contributed by Mario Carneiro, 7-Oct-2016.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  F/ x ( ps  /\  ch ) )
 
Theoremnf3and 1764 Deduction form of bound-variable hypothesis builder nf3an 1774. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  F/ x th )   =>    |-  ( ph  ->  F/ x ( ps  /\  ch  /\  th ) )
 
Theoremnfn 1765 If  x is not free in  ph, it is not free in  -.  ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |- 
 F/ x  -.  ph
 
Theoremnfal 1766 If  x is not free in  ph, it is not free in  A. y ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |- 
 F/ x A. y ph
 
Theoremnfex 1767 If  x is not free in  ph, it is not free in  E. y ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |- 
 F/ x E. y ph
 
Theoremnfnf 1768 If  x is not free in  ph, it is not free in  F/ y ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |- 
 F/ x F/ y ph
 
Theoremnfim 1769 If  x is not free in  ph and  ps, it is not free in  ( ph  ->  ps ). (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  ->  ps )
 
Theoremnfor 1770 If  x is not free in  ph and  ps, it is not free in  ( ph  \/  ps ). (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  \/  ps )
 
Theoremnfan 1771 If  x is not free in  ph and  ps, it is not free in  ( ph  /\  ps ). (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  /\  ps )
 
Theoremnfbi 1772 If  x is not free in  ph and  ps, it is not free in  ( ph  <->  ps ). (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  <->  ps )
 
Theoremnf3or 1773 If  x is not free in  ph,  ps, and  ch, it is not free in  ( ph  \/  ps  \/  ch ). (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   &    |-  F/ x ps   &    |-  F/ x ch   =>    |- 
 F/ x ( ph  \/  ps  \/  ch )
 
Theoremnf3an 1774 If  x is not free in  ph,  ps, and  ch, it is not free in  ( ph  /\  ps  /\  ch ). (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   &    |-  F/ x ps   &    |-  F/ x ch   =>    |- 
 F/ x ( ph  /\ 
 ps  /\  ch )
 
Theoremnfald 1775 If  x is not free in  ph, it is not free in  A. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y ps )
 
Theoremnfexd 1776 If  x is not free in  ph, it is not free in  E. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E. y ps )
 
Theoremnfa2 1777 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x A. y A. x ph
 
Theoremnfia1 1778 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ( A. x ph  ->  A. x ps )
 
Theoremmodal-b 1779 The analog in our "pure" predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.)
 |-  ( ph  ->  A. x  -.  A. x  -.  ph )
 
Theorem19.2g 1780 Theorem 19.2 of [Margaris] p. 89, generalized to use two set variables. (Contributed by O'Cat, 31-Mar-2008.)
 |-  ( A. x ph  ->  E. y ph )
 
Theorem19.3 1781 A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   =>    |-  ( A. x ph  <->  ph )
 
Theorem19.9t 1782 A closed version of 19.9 1783. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
 
Theorem19.9 1783 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.)
 |- 
 F/ x ph   =>    |-  ( E. x ph  <->  ph )
 
Theorem19.9d 1784 A deduction version of one direction of 19.9 1783. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |-  ( ps  ->  F/ x ph )   =>    |-  ( ps  ->  ( E. x ph  ->  ph )
 )
 
Theoremexcomim 1785 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |-  ( E. x E. y ph  ->  E. y E. x ph )
 
Theoremexcom 1786 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x E. y ph  <->  E. y E. x ph )
 
Theorem19.16 1787 Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph 
 <->  ps )  ->  ( ph 
 <-> 
 A. x ps )
 )
 
Theorem19.17 1788 Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ps   =>    |-  ( A. x ( ph  <->  ps )  ->  ( A. x ph  <->  ps ) )
 
Theorem19.19 1789 Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph 
 <->  ps )  ->  ( ph 
 <-> 
 E. x ps )
 )
 
Theorem19.21t 1790 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |-  ( F/ x ph  ->  ( A. x (
 ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )
 
Theorem19.21 1791 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 ) )
 
Theorem19.21-2 1792 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 ) )
 
Theoremstdpc5 1793 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 1645. 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.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph  ->  ps )  ->  ( ph  ->  A. x ps )
 )
 
Theorem19.21bi 1794 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.21bbi 1795 Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.)
 |-  ( ph  ->  A. x A. y ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.23t 1796 Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.)
 |-  ( F/ x ps  ->  ( A. x (
 ph  ->  ps )  <->  ( E. x ph 
 ->  ps ) ) )
 
Theorem19.23 1797 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 ) )
 
Theoremnf2 1798 An alternative definition of df-nf 1532, 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 ) )
 
Theoremnf3 1799 An alternative definition of df-nf 1532. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( F/ x ph  <->  A. x ( E. x ph 
 ->  ph ) )
 
Theoremnf4 1800 Variable  x is effectively not free in  ph iff  ph is always true or always false. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( F/ x ph  <->  ( A. x ph  \/  A. x  -.  ph ) )
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