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Theorem List for Metamath Proof Explorer - 33401-33500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembj-cbvexdv 33401* Version of cbvexd 1999 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  F/ y ph   &    |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. y ch )
 )
 
Theorembj-cbval2v 33402* Version of cbval2 2000 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  F/ z ph   &    |-  F/ w ph   &    |-  F/ x ps   &    |-  F/ y ps   &    |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x A. y ph  <->  A. z A. w ps )
 
Theorembj-cbvex2v 33403* Version of cbvex2 2001 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  F/ z ph   &    |-  F/ w ph   &    |-  F/ x ps   &    |-  F/ y ps   &    |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. x E. y ph  <->  E. z E. w ps )
 
Theorembj-cbval2vv 33404* Version of cbval2v 2003 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  (
 ( x  =  z 
 /\  y  =  w )  ->  ( ph  <->  ps ) )   =>    |-  ( A. x A. y ph  <->  A. z A. w ps )
 
Theorembj-cbvex2vv 33405* Version of cbvex2v 2004 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  (
 ( x  =  z 
 /\  y  =  w )  ->  ( ph  <->  ps ) )   =>    |-  ( E. x E. y ph  <->  E. z E. w ps )
 
Theorembj-cbvaldvav 33406* Version of cbvaldva 2005 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. y ch )
 )
 
Theorembj-cbvexdvav 33407* Version of cbvexdva 2006 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. y ch )
 )
 
Theorembj-cbvex4vv 33408* Version of cbvex4v 2007 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  (
 ( x  =  v 
 /\  y  =  u )  ->  ( ph  <->  ps ) )   &    |-  ( ( z  =  f  /\  w  =  g )  ->  ( ps 
 <->  ch ) )   =>    |-  ( E. x E. y E. z E. w ph  <->  E. v E. u E. f E. g ch )
 
Theorembj-equs4v 33409* Version of equs4 2008 with a dv condition, which does not require ax-13 1968 (neither ax-5 1680 nor ax-7 1739 nor ax-12 1803). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
 )
 
Theorembj-equsalv 33410* Version of equsal 2009 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
 
Theorembj-equsalhv 33411* Version of equsalh 2010 with a dv condition, which does not require ax-13 1968. Remark: this is the same as equsalhw 1892. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ( x  =  y  -> 
 ph )  <->  ps )
 
Theorembj-equsexv 33412* Version of equsex 2011 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
 
Theorembj-equsexhv 33413* Version of equsexh 2012 with a dv condition, which does not require ax-13 1968. Remark: the theorem axc9lem2 2013 has a dv version which is a simple consequence of ax5e 1682; the theorems nfeqf2 2014, dveeq2 2015, nfeqf1 2016, dveeq1 2017, nfeqf 2018, axc9 2019, ax13 2020, have dv versions which are simple consequences of ax-5 1680. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
 
Theorembj-axc11nlemv 33414* Version of axc11nlemOLD 2021 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  w  ->  A. y  y  =  x )
 
Theorembj-axc11nv 33415* Version of axc11n 2022 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theorembj-aecomsv 33416* Version of aecoms 2025 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ph )   =>    |-  ( A. y  y  =  x  ->  ph )
 
Theorembj-naecomsv 33417* Version of naecoms 2026 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ph )   =>    |-  ( -.  A. y  y  =  x  -> 
 ph )
 
Theorembj-axc11v 33418* Version of axc11 2027 with a dv condition, which does not require ax-13 1968. Remark: the following theorems (hbae 2028, nfae 2029, hbnae 2030, nfnae 2031, hbnaes 2032) would need to be totally unbundled to be proved without ax-13 1968, hence would be simple consequences of ax-5 1680 or nfv 1683. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( A. x ph 
 ->  A. y ph )
 )
 
Theorembj-aevlem1v 33419* Version of aevlem1 1886 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  ( A. z  z  =  w  ->  A. y  y  =  x )
 
Theorembj-axc16g 33420* Remove dependency on ax-13 1968 from axc16g 1887. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
 
Theorembj-aev 33421* Remove dependency on ax-13 1968 from aev 1890. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  A. z  w  =  v )
 
Theorembj-axc16 33422* Remove dependency on ax-13 1968 from axc16 1888. The same is doable for axc16i 2037 which has no interest. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
 
Theorembj-ax16nf 33423* Remove dependency on ax-13 1968 from ax16nf 1891. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  F/ z ph )
 
Theorembj-ax16gb 33424* Remove dependency on ax-13 1968 from ax16gb 1889. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  A. z ph )
 )
 
Theorembj-dral1v 33425* Version of dral1 2040 with a dv condition, which does not require ax-13 1968. Remark: the corresponding versions for dral2 2039 and drex2 2043 are instances of albidv 1689 and exbidv 1690 respectively. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. x ph  <->  A. y ps )
 )
 
Theorembj-drex1v 33426* Version of drex1 2042 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. y ps )
 )
 
Theorembj-drnf1v 33427* Version of drnf1 2044 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ y ps )
 )
 
Theorembj-drnf2v 33428* Version of drnf2 2045 with a dv condition, which does not require ax-13 1968. Could be labelled "nfbidv". (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps )
 )
 
Theorembj-axc15v 33429* Version of axc15 2058 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
 ) ) )
 
Theorembj-equs45fv 33430* Version of equs45f 2064 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
 |-  F/ y ph   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph )
 )
 
Theorembj-equs5v 33431* Version of equs5 2065 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) ) )
 
Theorembj-sb2v 33432* Version of sb2 2066 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
 
Theorembj-stdpc4v 33433* Version of stdpc4 2067 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  ( A. x ph  ->  [ y  /  x ] ph )
 
Theorembj-2stdpc4v 33434* Version of 2stdpc4 2068 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A. x A. y ph  ->  [ z  /  x ] [ w  /  y ] ph )
 
Theorembj-sb3v 33435* Version of sb3 2069 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )
 
Theorembj-sb4v 33436* Version of sb4 2070 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( [
 y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theorembj-sb4bv 33437* Version of sb4b 2071 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( [
 y  /  x ] ph 
 <-> 
 A. x ( x  =  y  ->  ph )
 ) )
 
Theorembj-hbsb2v 33438* Version of hbsb2 2072 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( [
 y  /  x ] ph  ->  A. x [ y  /  x ] ph )
 )
 
Theorembj-nfsb2v 33439* Version of nfsb2 2073 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  F/ x [ y  /  x ] ph )
 
Theorembj-hbsb2av 33440* Version of hbsb2a 2074 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
 |-  ( [ y  /  x ] A. y ph  ->  A. x [ y  /  x ] ph )
 
Theorembj-hbsb3v 33441* Version of hbsb3 2076 with a dv condition, which does not require ax-13 1968. (Remark: the unbundled version of nfs1 2077 is given by bj-nfs1v 33452.) (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( [ y  /  x ] ph  ->  A. x [
 y  /  x ] ph )
 
Theorembj-equsb1v 33442* Version of equsb1 2080 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
 |-  [ y  /  x ] x  =  y
 
Theorembj-cleljust 33443* Remove dependency on ax-13 1968 from cleljust 2082. (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.)
 |-  ( x  e.  y  <->  E. z ( z  =  x  /\  z  e.  y ) )
 
Theorembj-sbftv 33444* Version of sbft 2093 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
 
Theorembj-sbfv 33445* Version of sbf 2094 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  F/ x ph   =>    |-  ( [ y  /  x ] ph  <->  ph )
 
Theorembj-sbtv 33446* Version of sbt 2140 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  ph   =>    |- 
 [ y  /  x ] ph
 
Theorembj-ax12v 33447* Remove dependency on ax-13 1968 from ax12v 1804. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
 |-  ( x  =  y  ->  (
 ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theorembj-sb56 33448* Remove dependency on ax-13 1968 from sb56 2154. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
 |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
 
Theorembj-sb6 33449* Remove dependency on ax-13 1968 from sb6 2155. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
 |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 )
 
Theorembj-sb5 33450* Remove dependency on ax-13 1968 from sb5 2157. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
 |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )
 )
 
Theorembj-hbs1 33451* Remove dependency on ax-13 1968 from hbs1 2163. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  ( [ y  /  x ] ph  ->  A. x [
 y  /  x ] ph )
 
Theorembj-nfs1v 33452* Remove dependency on ax-13 1968 from nfs1v 2164. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  F/ x [ y  /  x ] ph
 
Theorembj-axext3 33453* Remove dependency on ax-13 1968 from axext3 2447. (Contributed by BJ, 12-Jul-2019.) (Proof modification is discouraged.)
 |-  ( A. z ( z  e.  x  <->  z  e.  y
 )  ->  x  =  y )
 
Theorembj-axext4 33454* Remove dependency on ax-13 1968 from axext4 2449. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
 |-  ( x  =  y  <->  A. z ( z  e.  x  <->  z  e.  y
 ) )
 
Theorembj-hbab1 33455* Remove dependency on ax-13 1968 from hbab1 2455. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  (
 y  e.  { x  |  ph }  ->  A. x  y  e.  { x  |  ph } )
 
Theorembj-nfsab1 33456* Remove dependency on ax-13 1968 from nfsab1 2456. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  F/ x  y  e.  { x  |  ph }
 
Theorembj-cleqh 33457* Remove dependency on ax-13 1968 from cleqh 2582. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
 |-  (
 y  e.  A  ->  A. x  y  e.  A )   &    |-  ( y  e.  B  ->  A. x  y  e.  B )   =>    |-  ( A  =  B  <->  A. x ( x  e.  A  <->  x  e.  B ) )
 
Theorembj-abeq2 33458* Remove dependency on ax-13 1968 from abeq2 2591. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A  =  { x  |  ph }  <->  A. x ( x  e.  A  <->  ph ) )
 
Theorembj-abeq1 33459* Remove dependency on ax-13 1968 from abeq1 2592. Remark: the theorems abeq2i 2594, abeq1i 2596, abeq2d 2593 do not use ax-11 1791 or ax-13 1968. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  ( { x  |  ph }  =  A 
 <-> 
 A. x ( ph  <->  x  e.  A ) )
 
Theorembj-abbi 33460 Remove dependency on ax-13 1968 from abbi 2598. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
 
Theorembj-abbi2i 33461* Remove dependency on ax-13 1968 from abbi2i 2600. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  ( x  e.  A  <->  ph )   =>    |-  A  =  { x  |  ph }
 
Theorembj-abbii 33462 Remove dependency on ax-13 1968 from abbii 2601. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  ( ph 
 <->  ps )   =>    |- 
 { x  |  ph }  =  { x  |  ps }
 
Theorembj-abbid 33463 Remove dependency on ax-13 1968 from abbid 2602. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
 
Theorembj-abbidv 33464* Remove dependency on ax-13 1968 from abbidv 2603. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
 
Theorembj-abbi2dv 33465* Remove dependency on ax-13 1968 from abbi2dv 2604. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  ( ph  ->  ( x  e.  A  <->  ps ) )   =>    |-  ( ph  ->  A  =  { x  |  ps } )
 
Theorembj-abbi1dv 33466* Remove dependency on ax-13 1968 from abbi1dv 2605. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  ( ph  ->  ( ps  <->  x  e.  A ) )   =>    |-  ( ph  ->  { x  |  ps }  =  A )
 
Theorembj-abid2 33467* Remove dependency on ax-13 1968 from abid2 2607. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  { x  |  x  e.  A }  =  A
 
Theorembj-clabel 33468* Remove dependency on ax-13 1968 from clabel 2613. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  ( { x  |  ph }  e.  A 
 <-> 
 E. y ( y  e.  A  /\  A. x ( x  e.  y  <->  ph ) ) )
 
Theorembj-sbab 33469* Remove dependency on ax-13 1968 from sbab 2614. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
 |-  ( x  =  y  ->  A  =  { z  |  [ y  /  x ] z  e.  A } )
 
Theorembj-nfab1 33470 Remove dependency on ax-13 1968 from nfab1 2631. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
 |-  F/_ x { x  |  ph }
 
Theorembj-vjust 33471 Remove dependency on ax-13 1968 from vjust 3114. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  { x  |  x  =  x }  =  { y  |  y  =  y }
 
Theorembj-cdeqab 33472* Remove dependency on ax-13 1968 from cdeqab 3321. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
 |- CondEq ( x  =  y  ->  ( ph 
 <->  ps ) )   =>    |- CondEq ( x  =  y  ->  { z  |  ph }  =  {
 z  |  ps }
 )
 
Theorembj-axrep1 33473* Remove dependency on ax-13 1968 from axrep1 4559. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  E. x ( E. y A. z
 ( ph  ->  z  =  y )  ->  A. z
 ( z  e.  x  <->  E. x ( x  e.  y  /\  ph )
 ) )
 
Theorembj-axrep2 33474* Remove dependency on ax-13 1968 from axrep2 4560. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  E. x ( E. y A. z
 ( ph  ->  z  =  y )  ->  A. z
 ( z  e.  x  <->  E. x ( x  e.  y  /\  A. y ph ) ) )
 
Theorembj-axrep3 33475* Remove dependency on ax-13 1968 from axrep3 4561. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  E. x ( E. y A. z
 ( ph  ->  z  =  y )  ->  A. z
 ( z  e.  x  <->  E. x ( x  e.  w  /\  A. y ph ) ) )
 
Theorembj-axrep4 33476* Remove dependency on ax-13 1968 from axrep4 4562. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  F/ z ph   =>    |-  ( A. x E. z A. y ( ph  ->  y  =  z ) 
 ->  E. z A. y
 ( y  e.  z  <->  E. x ( x  e.  w  /\  ph )
 ) )
 
Theorembj-axrep5 33477* Remove dependency on ax-13 1968 from axrep5 4563. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  F/ z ph   =>    |-  ( A. x ( x  e.  w  ->  E. z A. y (
 ph  ->  y  =  z ) )  ->  E. z A. y ( y  e.  z  <->  E. x ( x  e.  w  /\  ph )
 ) )
 
Theorembj-axsep 33478* Remove dependency on ax-13 1968 from axsep 4567. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
 )
 
Theorembj-nalset 33479* Remove dependency on ax-13 1968 from nalset 4584. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  -.  E. x A. y  y  e.  x
 
Theorembj-zfpow 33480* Remove dependency on ax-13 1968 from zfpow 4626. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  E. x A. y ( A. x ( x  e.  y  ->  x  e.  z ) 
 ->  y  e.  x )
 
Theorembj-el 33481* Remove dependency on ax-13 1968 from el 4629. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  E. y  x  e.  y
 
Theorembj-dtru 33482* Remove dependency on ax-13 1968 from dtru 4638. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
 |-  -.  A. x  x  =  y
 
Theorembj-axc16b 33483* Remove dependency on ax-13 1968 from axc16b 4639. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
 
Theorembj-eunex 33484 Remove dependency on ax-13 1968 from eunex 4640. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
 |-  ( E! x ph  ->  E. x  -.  ph )
 
Theorembj-dtrucor 33485* Remove dependency on ax-13 1968 from dtrucor 4680. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
 |-  x  =  y   =>    |-  x  =/=  y
 
Theorembj-dtrucor2v 33486* Version of dtrucor2 4681 with a dv condition, which does not require ax-13 1968 (nor ax-4 1612, ax-5 1680, ax-7 1739, ax-12 1803). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
 |-  ( x  =  y  ->  x  =/=  y )   =>    |-  ( ph  /\  -.  ph )
 
Theorembj-dvdemo1 33487* Remove dependency on ax-13 1968 from dvdemo1 4682. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
 |-  E. x ( x  =  y  ->  z  e.  x )
 
Theorembj-dvdemo2 33488* Remove dependency on ax-13 1968 from dvdemo2 4683. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
 |-  E. x ( x  =  y  ->  z  e.  x )
 
21.29.4.13  Strengthenings of theorems of the main part

Typically, these are biconditional versions of theorems in the main part which are formulated as implications. They could be added after said implication, or sometimes replace it (by "inlining" it).

This could also be done for hba1 1844, hbe1 1788, hbn1 1787, modal-5 1790.

 
Theorembj-sb3b 33489 Simplified definition of substitution when variables are distinct. This is to sb3 2069 what sb4b 2071 is to sb4 2070. Actually, one may keep only bj-sb3b 33489 and sb4b 2071 in the database, renaming them sb3 and sb4. (Contributed by BJ, 6-Oct-2018.)
 |-  ( -.  A. x  x  =  y  ->  ( [
 y  /  x ] ph 
 <-> 
 E. x ( x  =  y  /\  ph )
 ) )
 
21.29.4.14  Distinct var metavariables

The closed formula  A. x A. y
x  =  y approximately means that the var metavariables  x and  y represent the same variable vi. In a domain with at most one object, however, this formula is always true, hence the "approximately" in the previous sentence.

 
Theorembj-hbaeb2 33490 Biconditional version of a form of hbae 2028 with commuted quantifiers, not requiring ax-11 1791. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y 
 <-> 
 A. x A. z  x  =  y )
 
Theorembj-hbaeb 33491 Biconditional version of hbae 2028. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y 
 <-> 
 A. z A. x  x  =  y )
 
Theorembj-hbnaeb 33492 Biconditional version of hbnae 2030 (to replace it?). (Contributed by BJ, 6-Oct-2018.)
 |-  ( -.  A. x  x  =  y  <->  A. z  -.  A. x  x  =  y
 )
 
Theorembj-dvv 33493 A special instance of bj-hbaeb2 33490. A lemma for distinct var metavariables. Note that the right-hand side is a closed formula (a sentence). (Contributed by BJ, 6-Oct-2018.)
 |-  ( A. x  x  =  y 
 <-> 
 A. x A. y  x  =  y )
 
21.29.4.15  Around ~ equsal

As a rule of thumb, if a theorem of the form  |-  ( ph  <->  ps ) =>  |-  ( ch 
<->  th ) is in the database, and the "more precise" theorems  |-  ( ph  ->  ps ) =>  |-  ( ch  ->  th ) and  |-  ( ps 
->  ph ) =>  |-  ( th  ->  ch ) also hold (see bj-bisym 33278), then they should be added to the database. The present case is similar. Similar additions can be done regarding equsex 2011 (and equsalh 2010 and equsexh 2012). Even if only one of these two theorems holds, it should be added to the database.

 
Theorembj-equsal1t 33494 Duplication of wl-equsal1t 29599, with shorter proof. Note: wl-equsalcom 29600 is also interesting. (Contributed by BJ, 6-Oct-2018.)
 |-  ( F/ x ph  ->  ( A. x ( x  =  y  ->  ph )  <->  ph ) )
 
Theorembj-equsal1ti 33495 Inference associated with bj-equsal1t 33494. (Contributed by BJ, 30-Sep-2018.)
 |-  F/ x ph   =>    |-  ( A. x ( x  =  y  ->  ph )  <->  ph )
 
Theorembj-equsal1 33496 One direction of equsal 2009. (Contributed by BJ, 30-Sep-2018.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( A. x ( x  =  y  -> 
 ph )  ->  ps )
 
Theorembj-equsal2 33497 One direction of equsal 2009. (Contributed by BJ, 30-Sep-2018.)
 |-  F/ x ph   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( ph  ->  A. x ( x  =  y  ->  ps )
 )
 
Theorembj-equsal 33498 Shorter proof of equsal 2009. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2009, but "min */exc equsal" is ok. (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
 
21.29.4.16  Some Principia Mathematica proofs

References are made to the second edition (1927, reprinted 1963) of Principia Mathematica, Vol. 1. Theorems are referred to in the form "PM*xx.xx".

 
Theoremstdpc5t 33499 Closed form of stdpc5 1855. (Possible to place it before 19.21t 1852 and use it to prove 19.21t 1852). (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
 |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  ->  ( ph  ->  A. x ps )
 ) )
 
Theorembj-stdpc5 33500 More direct proof of stdpc5 1855. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
 |-  F/ x ph   =>    |-  ( A. x (
 ph  ->  ps )  ->  ( ph  ->  A. x ps )
 )
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