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Theorem List for Metamath Proof Explorer - 28801-28900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremdvelimvNEW7 28801* Similar to dvelim 2051 with first hypothesis replaced by distinct variable condition. (Contributed by NM, 25-Jul-2015.)
 |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdveeq2NEW7 28802* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.)
 |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
 )
 
Theoremdveeq1NEW7 28803* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )
 
Theoremdveel1NEW7 28804* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( y  e.  z  ->  A. x  y  e.  z )
 )
 
Theoremdveel2NEW7 28805* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( z  e.  y  ->  A. x  z  e.  y )
 )
 
TheoremdvelimwAUX7 28806* Weaker version of dvelim 2051. (Contributed by NM, 23-Nov-1994.)
 |-  ( ph  ->  A. x ph )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremax9NEW7 28807 Theorem showing that ax-9 1661 follows from the weaker version ax9v 1662. (Even though this theorem depends on ax-9 1661, all references of ax-9 1661 are made via ax9v 1662. An earlier version stated ax9v 1662 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-9 1661 so that all proofs can be traced back to ax9v 1662. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.)

 |-  -.  A. x  -.  x  =  y
 
Theoremax9oNEW7 28808 Show that the original axiom ax-9o 2173 can be derived from ax9 1942 and others. See ax9from9o 2183 for the rederivation of ax9 1942 from ax-9o 2173.

Normally, ax9o 1943 should be used rather than ax-9o 2173, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)

 |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )
 
Theorema9eNEW7 28809 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 1563 through ax-14 1721 and ax-17 1623, all axioms other than ax9 1942 are believed to be theorems of free logic, although the system without ax9 1942 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.)
 |-  E. x  x  =  y
 
Theoremax10lem4NEW7 28810* Lemma for ax10 1984. Change bound variable. (Contributed by NM, 8-Jul-2016.)
 |-  ( A. x  x  =  w  ->  A. y  y  =  x )
 
Theoremax10lem5NEW7 28811* Lemma for ax10 1984. Change free and bound variables. (Contributed by NM, 22-Jul-2015.)
 |-  ( A. z  z  =  w  ->  A. y  y  =  x )
 
Theoremax10NEW7 28812 Derive set.mm's original ax-10 2175 from others. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
TheoremaecomNEW7 28813 Commutation law for identical variable specifiers. The antecedent and consequent are true when  x and  y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
TheoremaecomsNEW7 28814 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ph )   =>    |-  ( A. y  y  =  x  ->  ph )
 
Theoremax10oNEW7 28815 Show that ax-10o 2174 can be derived from ax-10 2175 in the form of ax10 1984. Normally, ax10o 1993 should be used rather than ax-10o 2174, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( A. x ph 
 ->  A. y ph )
 )
 
Theoremhba1eAUX7 28816 Special case of hbae 1999 not requiring ax-7 1741. (Contributed by NM, 12-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
 
TheoremhbaewAUX7 28817* Weak version of hbae 1999 not requiring ax-7 1741. See hbaew2AUX7 28818 and hbaew3AUX7 28861 for versions with different distinct variable requirements. (Contributed by NM, 10-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
Theoremhbaew2AUX7 28818* Weak version of hbae 1999 not requiring ax-7 1741. Different distinct variable requirements from hbaewAUX7 28817. (Contributed by NM, 30-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
TheoremnfaewAUX7 28819* Weak version of nfae 2000 not requiring ax-7 1741. (Contributed by NM, 10-Oct-2017.)
 |-  F/ z A. x  x  =  y
 
TheoremhbnaewAUX7 28820* Weak version of hbnae 2001 not requiring ax-7 1741. (Contributed by NM, 10-Oct-2017.)
 |-  ( -.  A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
 
TheoremnfnaewAUX7 28821* Weak version of nfnae 2002 not requiring ax-7 1741. (Contributed by NM, 27-Oct-2017.)
 |-  F/ z  -.  A. x  x  =  y
 
Theoremnfaew2AUX7 28822* Weak version of nfae 2000 not requiring ax-7 1741. (Contributed by NM, 25-Nov-2017.)
 |-  F/ z A. x  x  =  y
 
Theoremhbnaew2AUX7 28823* Weak version of hbnae 2001 not requiring ax-7 1741. (Contributed by NM, 25-Nov-2017.)
 |-  ( -.  A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
 
Theoremnfnaew2AUX7 28824* Weak version of nfnae 2002 not requiring ax-7 1741. (Contributed by NM, 25-Nov-2017.)
 |-  F/ z  -.  A. x  x  =  y
 
TheoremnfeqfNEW7 28825 A variable is effectively not free in an equality if it is not either of the involved variables.  F/ version of ax-12o 2177. (Contributed by Mario Carneiro, 6-Oct-2016.)
 |-  (
 ( -.  A. z  z  =  x  /\  -. 
 A. z  z  =  y )  ->  F/ z  x  =  y )
 
TheoremequsalNEW7 28826 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.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
 
TheoremequsalihAUX7 28827 One direction of equsalhNEW7 28828 with weaker hypothesis. TO DO: Delete if not used. (Contributed by NM, 13-Nov-2017.)
 |-  ( x  =  y  ->  (
 ph  ->  A. x ps )
 )   =>    |-  ( A. x ( x  =  y  ->  ph )  ->  ps )
 
TheoremequsalhNEW7 28828 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ( x  =  y  -> 
 ph )  <->  ps )
 
TheoremequsexNEW7 28829 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
 
TheoremequsexhNEW7 28830 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
 
TheoremdvelimhvAUX7 28831* Weak version of dvelimh 2005 not requiring ax-7 1741. (Contributed by NM, 10-Oct-2017.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdral1NEW7 28832 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 )
 )
 
Theoremdrex1NEW7 28833 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. x ph  <->  E. y ps )
 )
 
Theoremdrnf1NEW7 28834 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 )
 )
 
Theoremdral2wAUX7 28835* Weak version of dral2 2007 not requiring ax-7 1741. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. z ph  <->  A. z ps )
 )
 
Theoremdrex2wAUX7 28836* Weak version of drex2 2009 not requiring ax-7 1741. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. z ph  <->  E. z ps )
 )
 
Theoremdrnf2wAUX7 28837* Weak version of drnf2 2011 not requiring ax-7 1741. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps )
 )
 
Theoremdral2w2AUX7 28838* Weak version of dral2 2007 not requiring ax-7 1741. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. z ph  <->  A. z ps )
 )
 
Theoremdrex2w2AUX7 28839* Weak version of drex2 2009 not requiring ax-7 1741. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. z ph  <->  E. z ps )
 )
 
Theoremdrnf2w2AUX7 28840* Weak version of drnf2 2011 not requiring ax-7 1741. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps )
 )
 
Theoremdral2w3AUX7 28841 Weak version of dral2 2007 not requiring ax-7 1741. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. x ph  <->  A. x ps )
 )
 
Theoremdrex2w3AUX7 28842 Weak version of drex2 2009 not requiring ax-7 1741. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. x ps )
 )
 
Theoremdrnf2w3AUX7 28843 Weak version of drnf2 2011 not requiring ax-7 1741. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ x ps )
 )
 
TheoremexdistrfNEW7 28844 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 Mario Carneiro, 20-Mar-2013.) (Revised by NM, 25-Nov-2017.) Revised to prove from ax-7v 28781 instead of ax-7 1741.
 |-  ( -.  A. x  x  =  y  ->  F/ y ph )   =>    |-  ( E. x E. y ( ph  /\  ps )  ->  E. x ( ph  /\ 
 E. y ps )
 )
 
Theoremdrsb1NEW7 28845 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 ) )
 
TheoremspimtNEW7 28846 Closed theorem form of spim 1948. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.)
 |-  (
 ( F/ x ps  /\ 
 A. x ( x  =  y  ->  ( ph  ->  ps ) ) ) 
 ->  ( A. x ph  ->  ps ) )
 
TheoremspimNEW7 28847 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1948 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.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
TheoremspimeNEW7 28848 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.)
 |-  F/ x ph   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( ph  ->  E. x ps )
 
TheoremspimedNEW7 28849 Deduction version of spime 1950. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  ( ch  ->  F/ x ph )   &    |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( ch  ->  ( ph  ->  E. x ps )
 )
 
Theoremcbv1hwAUX7 28850* Weak version of cbv1h 2015 not requiring ax-7 1741. (Contributed by NM, 28-Oct-2017.)
 |-  ( 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 ) )
 
Theoremcbv1wAUX7 28851* Weak version of cbv1 2016 not requiring ax-7 1741. (Contributed by NM, 28-Oct-2017.)
 |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch )
 )
 
Theoremcbv2hwAUX7 28852* Weak version of cbv2h 2017 not requiring ax-7 1741. (Contributed by NM, 28-Oct-2017.)
 |-  ( 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 )
 )
 
Theoremcbv2wAUX7 28853* Weak version of cbv2 2018 not requiring ax-7 1741. (Contributed by NM, 28-Oct-2017.)
 |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch )
 )
 
Theoremcbv3wAUX7 28854* Weak version of cbv3 2019 not requiring ax-7 1741. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremcbv3hwAUX7 28855* Weak version of cbv3h 2020 not requiring ax-7 1741. (Contributed by NM, 28-Oct-2017.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
TheoremcbvalwwAUX7 28856* Weak version of cbval 2021 not requiring ax-7 1741. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
TheoremcbvexwAUX7 28857* Weak version of cbvex 2022 not requiring ax-7 1741. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
TheoremspimvNEW7 28858* A version of spim 1948 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 )
 
TheoremaevwAUX7 28859* Weak version of aev 2027 not requiring ax-7 1741. (Contributed by NM, 28-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. z  w  =  v )
 
TheoremaevNEW7 28860* A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.)
 |-  ( A. x  x  =  y  ->  A. z  w  =  v )
 
Theoremhbaew3AUX7 28861* Weak version of hbae 1999 not requiring ax-7 1741. Has different distinct variable requirements from hbaewAUX7 28817. (Contributed by NM, 30-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
Theoremnfaew3AUX7 28862* Weak version of nfae 2000 not requiring ax-7 1741. (Contributed by NM, 25-Nov-2017.)
 |-  F/ z A. x  x  =  y
 
Theoremnfnaew3AUX7 28863* Weak version of nfnae 2002 not requiring ax-7 1741. (Contributed by NM, 25-Nov-2017.)
 |-  F/ z  -.  A. x  x  =  y
 
TheoremequviniNEW7 28864 A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require  z to be distinct from  x and  y (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) )
 
TheoremequveliNEW7 28865 A variable elimination law for equality with no distinct variable requirements. (Compare equviniNEW7 28864.) (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Revised by NM, 25-Nov-2017.) Revised to prove from ax-7v 28781 instead of ax-7 1741.
 |-  ( A. z ( z  =  x  <->  z  =  y
 )  ->  x  =  y )
 
TheoremequvinNEW7 28866* 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 ) )
 
Theoremax11v2NEW7 28867* Recovery of ax-11o 2176 from ax11v 2130. This proof uses ax-10 2175 and ax-11 1753. TODO: figure out if this is useful, or if it should be simplified or eliminated. (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 ) ) ) )
 
Theoremax11a2NEW7 28868* Derive ax-11o 2176 from a hypothesis in the form of ax-11 1753. ax-10 2175 and ax-11 1753 are used by the proof, but not ax-10o 2174 or ax-11o 2176. TODO: figure out if this is useful, or if it should be simplified or eliminated. (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 ) ) ) )
 
Theoremax11oNEW7 28869 Derivation of set.mm's original ax-11o 2176 from ax-10 2175 and the shorter ax-11 1753 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 2179 or ax-17 1623 (given all of the original and new versions of sp 1755 through ax-15 2178).

Another open problem is whether this theorem can be proved without relying on ax12o 1966.

Theorem ax11 2190 shows the reverse derivation of ax-11 1753 from ax-11o 2176.

Normally, ax11o 2030 should be used rather than ax-11o 2176, 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 )
 ) ) )
 
Theoremequs4NEW7 28870 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 )
 )
 
Theoremequs5NEW7 28871 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 )
 ) )
 
Theoremequs5bAUX7 28872 Lemma used in proofs of substitution properties. (Contributed by NM, 27-Oct-2017.)
 |-  ( -.  A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremax15NEW7 28873 Axiom ax-15 2178 is redundant if we assume ax-17 1623. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.

Note that  w is a dummy variable introduced in the proof. On the web page, it is implicitly assumed to be distinct from all other variables. (This is made explicit in the database file set.mm). Its purpose is to satisfy the distinct variable requirements of dveel2 2054 and ax-17 1623. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements. (Contributed by NM, 29-Jun-1995.)

 |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y 
 ->  ( x  e.  y  ->  A. z  x  e.  y ) ) )
 
Theoremsb2NEW7 28874 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
 
Theoremequsb1NEW7 28875 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
 |-  [ y  /  x ] x  =  y
 
Theoremequsb2NEW7 28876 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
 |-  [ y  /  x ] y  =  x
 
TheoremsbiedNEW7 28877 Conversion of implicit substitution to explicit substitution (deduction version of sbie 2072). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |-  F/ x ph   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch ) )
 
TheoremsbieNEW7 28878 Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( [ y  /  x ] ph  <->  ps )
 
Theoremhbsb2aNEW7 28879 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 )
 
Theoremhbsb2eNEW7 28880 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 )
 
Theoremhbsb3NEW7 28881 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 )
 
Theoremnfs1NEW7 28882 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
 
Theorema16gNEW7 28883* Generalization of ax16 2079. (Contributed by NM, 25-Jul-2015.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
 
Theorema16gbNEW7 28884* A generalization of axiom ax-16 2179. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  A. z ph )
 )
 
Theorema16nfwAUX7 28885* Weak version of a16nf 2085 not requiring ax-7 1741. (Contributed by NM, 10-Oct-2017.)
 |-  ( A. x  x  =  y  ->  F/ z ph )
 
Theoremax16NEW7 28886* Proof of older axiom ax-16 2179. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
 
Theorema16nfNEW7 28887* If dtru 4332 is false, then there is only one element in the universe, so everything satisfies  F/. (Contributed by Mario Carneiro, 7-Oct-2016.) (Revised by NM, 25-Nov-2017.) Revised to prove from ax-7v 28781 instead of ax-7 1741.
 |-  ( A. x  x  =  y  ->  F/ z ph )
 
Theoremax16iNEW7 28888* Inference with ax16 2079 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)
 |-  ( x  =  z  ->  (
 ph 
 <->  ps ) )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
 
Theoremsb4NEW7 28889 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 )
 ) )
 
Theoremsb4bNEW7 28890 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 )
 ) )
 
Theoremhbsb2NEW7 28891 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( [
 y  /  x ] ph  ->  A. x [ y  /  x ] ph )
 )
 
Theoremstdpc4NEW7 28892 The specialization axiom of standard predicate calculus. It states that if a statement  ph holds for all  x, then it also holds for the specific case of  y (properly) substituted for  x. Translated to traditional notation, it can be read: " A. x ph ( x )  ->  ph ( y ), provided that  y is free for  x in  ph (
x )." Axiom 4 of [Mendelson] p. 69. See also spsbc 3117 and rspsbc 3183. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  ->  [ y  /  x ] ph )
 
TheoremsbftNEW7 28893 Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
 
TheoremsbfNEW7 28894 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |-  F/ x ph   =>    |-  ( [ y  /  x ] ph  <->  ph )
 
Theoremsbequ5wAUX7 28895* Weak version of sbequ5 2065 not requiring ax-7 1741. (Contributed by NM, 27-Oct-2017.)
 |-  ( [ w  /  z ] A. x  x  =  y  <->  A. x  x  =  y )
 
TheoremsbhNEW7 28896 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( [ y  /  x ] ph  <->  ph )
 
Theoremsbf2NEW7 28897 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
 |-  ( [ y  /  x ] A. x ph  <->  A. x ph )
 
Theoremnfsb2NEW7 28898 Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.)
 |-  ( -.  A. x  x  =  y  ->  F/ x [ y  /  x ] ph )
 
TheoremsbnNEW7 28899 Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
 
Theoremsbi1NEW7 28900 Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  ->  ps )  ->  ( [
 y  /  x ] ph  ->  [ y  /  x ] ps ) )
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