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Theorem List for Metamath Proof Explorer - 28001-28100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembnj258 28001  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ph  /\ 
 ps  /\  th )  /\  ch ) )
 
Theorembnj268 28002  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  ch  /\ 
 ps  /\  th )
 )
 
Theorembnj290 28003  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  ch  /\ 
 th  /\  ps )
 )
 
Theorembnj291 28004  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ph  /\ 
 ch  /\  th )  /\  ps ) )
 
Theorembnj312 28005  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ps  /\  ph 
 /\  ch  /\  th )
 )
 
Theorembnj334 28006  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ch  /\  ph 
 /\  ps  /\  th )
 )
 
Theorembnj345 28007  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( th  /\  ph 
 /\  ps  /\  ch )
 )
 
Theorembnj422 28008  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ch  /\  th 
 /\  ph  /\  ps )
 )
 
Theorembnj432 28009  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ch 
 /\  th )  /\  ( ph  /\  ps ) ) )
 
Theorembnj446 28010  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ps 
 /\  ch  /\  th )  /\  ph ) )
 
Theorembnj21 28011* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { x  e.  A  |  ph }   =>    |-  B  C_  A
 
Theorembnj23 28012* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  B  =  { x  e.  A  |  -.  ph }   =>    |-  ( A. z  e.  B  -.  z R y  ->  A. w  e.  A  ( w R y  ->  [. w  /  x ]. ph ) )
 
Theorembnj31 28013 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  E. x  e.  A  ps )   &    |-  ( ps  ->  ch )   =>    |-  ( ph  ->  E. x  e.  A  ch )
 
Theorembnj62 28014* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( [. z  /  x ]. x  Fn  A  <->  z  Fn  A )
 
Theorembnj89 28015* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  Z  e.  _V   =>    |-  ( [. Z  /  y ]. E! x ph  <->  E! x [. Z  /  y ]. ph )
 
Theorembnj90 28016* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  Y  e.  _V   =>    |-  ( [. Y  /  x ]. z  Fn  x  <->  z  Fn  Y )
 
Theorembnj101 28017 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  E. x ph   &    |-  ( ph  ->  ps )   =>    |-  E. x ps
 
Theorembnj105 28018 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  1o  e.  _V
 
Theorembnj115 28019 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <-> 
 A. n  e.  D  ( ta  ->  th )
 )   =>    |-  ( et  <->  A. n ( ( n  e.  D  /\  ta )  ->  th )
 )
 
Theorembnj132 28020* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <-> 
 E. x ( ps 
 ->  ch ) )   =>    |-  ( ph  <->  ( ps  ->  E. x ch ) )
 
Theorembnj133 28021 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <-> 
 E. x ps )   &    |-  ( ch 
 <->  ps )   =>    |-  ( ph  <->  E. x ch )
 
Theorembnj142 28022 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  =  <. A ,  ( F `  A )
 >. ) )
 
Theorembnj145 28023 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  ( F `  A )  e.  _V   =>    |-  ( F  Fn  { A }  ->  F  =  { <. A ,  ( F `  A ) >. } )
 
Theorembnj156 28024 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ze0  <->  ( f  Fn  1o  /\  ph'  /\  ps' ) )   &    |-  ( ze1  <->  [. g  /  f ]. ze0 )   &    |-  ( ph1  <->  [. g  /  f ]. ph' )   &    |-  ( ps1  <->  [. g  /  f ]. ps' )   =>    |-  ( ze1  <->  ( g  Fn 
 1o  /\  ph1  /\  ps1 ) )
 
Theorembnj158 28025* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   =>    |-  ( m  e.  D  ->  E. p  e.  om  m  =  suc  p )
 
Theorembnj168 28026* First-order logic and set theory. Revised to remove dependence on ax-reg 7302. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by NM, 21-Dec-2016.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   =>    |-  ( ( n  =/= 
 1o  /\  n  e.  D )  ->  E. m  e.  D  n  =  suc  m )
 
Theorembnj206 28027 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph'  <->  [. M  /  n ]. ph )   &    |-  ( ps'  <->  [. M  /  n ].
 ps )   &    |-  ( ch'  <->  [. M  /  n ].
 ch )   &    |-  M  e.  _V   =>    |-  ( [. M  /  n ]. ( ph  /\  ps  /\ 
 ch )  <->  ( ph'  /\  ps'  /\  ch' ) )
 
Theorembnj216 28028 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  e.  _V   =>    |-  ( A  =  suc  B 
 ->  B  e.  A )
 
Theorembnj219 28029 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( n  =  suc  m  ->  m  _E  n )
 
Theorembnj226 28030* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  C_  C   =>    |-  U_ x  e.  A  B  C_  C
 
Theorembnj228 28031 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.)
 |-  ( ph 
 <-> 
 A. x  e.  A  ps )   =>    |-  ( ( x  e.  A  /\  ph )  ->  ps )
 
Theorembnj519 28032 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( B  e.  _V  ->  Fun  { <. A ,  B >. } )
 
Theorembnj521 28033 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( A  i^i  { A }
 )  =  (/)
 
Theorembnj524 28034 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ps )   &    |-  A  e.  _V   =>    |-  ( [. A  /  x ].
 ph 
 <-> 
 [. A  /  x ].
 ps )
 
Theorembnj525 28035* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( [. A  /  x ]. ph  <->  ph )
 
Theorembnj534 28036* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch  ->  ( E. x ph 
 /\  ps ) )   =>    |-  ( ch  ->  E. x ( ph  /\  ps ) )
 
Theorembnj538 28037* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( [. A  /  y ]. A. x  e.  B  ph  <->  A. x  e.  B  [. A  /  y ]. ph )
 
Theorembnj529 28038 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   =>    |-  ( M  e.  D  -> 
 (/)  e.  M )
 
Theorembnj551 28039 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( m  =  suc  p 
 /\  m  =  suc  i )  ->  p  =  i )
 
Theorembnj563 28040 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( rh  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  =/= 
 suc  i ) )   =>    |-  ( ( et  /\  rh )  ->  suc  i  e.  m )
 
Theorembnj564 28041 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ta 
 <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   =>    |-  ( ta  ->  dom  f  =  m )
 
Theorembnj593 28042 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  E. x ps )   &    |-  ( ps  ->  ch )   =>    |-  ( ph  ->  E. x ch )
 
Theorembnj596 28043 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  E. x ps )   =>    |-  ( ph  ->  E. x ( ph  /\ 
 ps ) )
 
Theorembnj610 28044* Pass from equality ( x  =  A) to substitution ( [. A  /  x ].) without the distinct variable restriction ($d  A  x). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  ( ph  <->  ps' ) )   &    |-  ( y  =  A  ->  ( ps'  <->  ps ) )   =>    |-  ( [. A  /  x ]. ph  <->  ps )
 
Theorembnj642 28045  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ph )
 
Theorembnj643 28046  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ps )
 
Theorembnj645 28047  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  th )
 
Theorembnj658 28048  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ( ph  /\  ps  /\  ch ) )
 
Theorembnj667 28049  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ( ps  /\  ch  /\  th ) )
 
Theorembnj705 28050  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 ch  /\  th )  ->  ta )
 
Theorembnj706 28051  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 ch  /\  th )  ->  ta )
 
Theorembnj707 28052  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 ch  /\  th )  ->  ta )
 
Theorembnj708 28053  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( th  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 ch  /\  th )  ->  ta )
 
Theorembnj721 28054  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  ta )   =>    |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ta )
 
Theorembnj832 28055  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps ) )   &    |-  ( ph  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj833 28056  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps ) )   &    |-  ( ps  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj835 28057  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps  /\ 
 ch ) )   &    |-  ( ph  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj836 28058  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps  /\ 
 ch ) )   &    |-  ( ps  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj837 28059  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps  /\ 
 ch ) )   &    |-  ( ch  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj769 28060  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps  /\ 
 ch  /\  th )
 )   &    |-  ( ph  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj770 28061  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps  /\ 
 ch  /\  th )
 )   &    |-  ( ps  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj771 28062  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps  /\ 
 ch  /\  th )
 )   &    |-  ( ch  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj887 28063  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ph' )   &    |-  ( ps  <->  ps' )   &    |-  ( ch  <->  ch' )   &    |-  ( th  <->  th' )   =>    |-  ( ( ph  /\  ps  /\ 
 ch  /\  th )  <->  ( ph'  /\  ps'  /\  ch'  /\  th' ) )
 
Theorembnj918 28064 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  G  e.  _V
 
Theorembnj919 28065* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( n  e.  D  /\  F  Fn  n  /\  ph 
 /\  ps ) )   &    |-  ( ph'  <->  [. P  /  n ]. ph )   &    |-  ( ps'  <->  [. P  /  n ].
 ps )   &    |-  ( ch'  <->  [. P  /  n ].
 ch )   &    |-  P  e.  _V   =>    |-  ( ch'  <->  ( P  e.  D  /\  F  Fn  P  /\  ph'  /\  ps' ) )
 
Theorembnj923 28066 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   =>    |-  ( n  e.  D  ->  n  e.  om )
 
Theorembnj926 28067 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ( ps  <->  ph ) )  ->  ps )
 
Theorembnj927 28068 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  C  e.  _V   =>    |-  (
 ( p  =  suc  n 
 /\  f  Fn  n )  ->  G  Fn  p )
 
Theorembnj930 28069 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  F  Fn  A )   =>    |-  ( ph  ->  Fun  F )
 
Theorembnj931 28070 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  =  ( B  u.  C )   =>    |-  B  C_  A
 
Theorembnj937 28071* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  E. x ps )   =>    |-  ( ph  ->  ps )
 
Theorembnj941 28072 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  ( C  e.  _V  ->  ( ( p  = 
 suc  n  /\  f  Fn  n )  ->  G  Fn  p ) )
 
Theorembnj945 28073 Technical lemma for bnj69 28308. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  ( ( C  e.  _V 
 /\  f  Fn  n  /\  p  =  suc  n 
 /\  A  e.  n )  ->  ( G `  A )  =  (
 f `  A )
 )
 
Theorembnj946 28074 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <-> 
 A. x  e.  A  ps )   =>    |-  ( ph  <->  A. x ( x  e.  A  ->  ps )
 )
 
Theorembnj951 28075  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ta  ->  ph )   &    |-  ( ta  ->  ps )   &    |-  ( ta  ->  ch )   &    |-  ( ta  ->  th )   =>    |-  ( ta  ->  ( ph  /\  ps  /\  ch  /\ 
 th ) )
 
Theorembnj956 28076 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( A  =  B  ->  A. x  A  =  B )   =>    |-  ( A  =  B  -> 
 U_ x  e.  A  C  =  U_ x  e.  B  C )
 
Theorembnj976 28077* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( N  e.  D  /\  f  Fn  N  /\  ph  /\  ps )
 )   &    |-  ( ph'  <->  [. G  /  f ]. ph )   &    |-  ( ps'  <->  [. G  /  f ]. ps )   &    |-  ( ch'  <->  [. G  /  f ]. ch )   &    |-  G  e.  _V   =>    |-  ( ch'  <->  ( N  e.  D  /\  G  Fn  N  /\  ph'  /\  ps' ) )
 
Theorembnj982 28078 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( ch  ->  A. x ch )   &    |-  ( th  ->  A. x th )   =>    |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  A. x ( ph  /\  ps  /\  ch 
 /\  th ) )
 
Theorembnj1019 28079* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( E. p ( th  /\  ch 
 /\  ta  /\  et )  <->  ( th  /\  ch  /\  et  /\  E. p ta ) )
 
Theorembnj1023 28080 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  E. x ( ph  ->  ps )   &    |-  ( ps  ->  ch )   =>    |- 
 E. x ( ph  ->  ch )
 
Theorembnj1095 28081 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <-> 
 A. x  e.  A  ps )   =>    |-  ( ph  ->  A. x ph )
 
Theorembnj1096 28082* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps 
 <->  ( ch  /\  th  /\ 
 ta  /\  ph ) )   =>    |-  ( ps  ->  A. x ps )
 
Theorembnj1098 28083* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   =>    |- 
 E. j ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  ->  (
 j  e.  n  /\  i  =  suc  j ) )
 
Theorembnj1101 28084 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  E. x ( ph  ->  ps )   &    |-  ( ch  ->  ph )   =>    |- 
 E. x ( ch 
 ->  ps )
 
Theorembnj1113 28085* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( A  =  B  ->  C  =  D )   =>    |-  ( A  =  B  ->  U_ x  e.  C  E  =  U_ x  e.  D  E )
 
Theorembnj1109 28086 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  E. x ( ( A  =/=  B 
 /\  ph )  ->  ps )   &    |-  (
 ( A  =  B  /\  ph )  ->  ps )   =>    |-  E. x ( ph  ->  ps )
 
Theorembnj1131 28087 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  E. x ph   =>    |-  ph
 
Theorembnj1138 28088 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  =  ( B  u.  C )   =>    |-  ( X  e.  A  <->  ( X  e.  B  \/  X  e.  C )
 )
 
Theorembnj1142 28089 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ( x  e.  A  ->  ps )
 )   =>    |-  ( ph  ->  A. x  e.  A  ps )
 
Theorembnj1143 28090* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  U_ x  e.  A  B  C_  B
 
Theorembnj1146 28091* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 y  e.  A  ->  A. x  y  e.  A )   =>    |-  U_ x  e.  A  B  C_  B
 
Theorembnj1149 28092 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   =>    |-  ( ph  ->  ( A  u.  B )  e. 
 _V )
 
Theorembnj1153 28093 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  X  e.  ( A  i^i  B ) )
 
Theorembnj1185 28094* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  E. z  e.  B  A. w  e.  B  -.  w R z )   =>    |-  ( ph  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
 
Theorembnj1196 28095 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  E. x  e.  A  ps )   =>    |-  ( ph  ->  E. x ( x  e.  A  /\  ps ) )
 
Theorembnj1198 28096 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  E. x ps )   &    |-  ( ps'  <->  ps )   =>    |-  ( ph  ->  E. x ps' )
 
Theorembnj1209 28097* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch  ->  E. x  e.  B  ph )   &    |-  ( th  <->  ( ch  /\  x  e.  B  /\  ph ) )   =>    |-  ( ch  ->  E. x th )
 
Theorembnj1211 28098 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  A. x  e.  A  ps )   =>    |-  ( ph  ->  A. x ( x  e.  A  ->  ps ) )
 
Theorembnj1213 28099 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  C_  B   &    |-  ( th  ->  x  e.  A )   =>    |-  ( th  ->  x  e.  B )
 
Theorembnj1212 28100* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { x  e.  A  |  ph }   &    |-  ( th  <->  ( ch  /\  x  e.  B  /\  ta ) )   =>    |-  ( th  ->  x  e.  A )
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