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Theorem List for Metamath Proof Explorer - 27401-27500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-bnj17 27401 Define the 4-way conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ph  /\ 
 ps  /\  ch )  /\  th ) )
 
Syntaxc-bnj14 27402 Extend class notation with the function giving: the class of all elements of  A that are "smaller" than  X according to  R. (New usage is discouraged.)
 class  pred ( X ,  A ,  R )
 
Definitiondf-bnj14 27403* Define the function giving: the class of all elements of  A that are "smaller" than  X according to  R. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  pred ( X ,  A ,  R )  =  {
 y  e.  A  |  y R X }
 
Syntaxw-bnj13 27404 Extend wff notation with the following predicate:  R is set-like on  A. (New usage is discouraged.)
 wff  R  Se  A
 
Definitiondf-bnj13 27405* Define the following predicate:  R is set-like on  A. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( R  Se  A  <->  A. x  e.  A  pred ( x ,  A ,  R )  e.  _V )
 
Syntaxw-bnj15 27406 Extend wff notation with the following predicate:  R is both well-founded and set-like on 
A. (New usage is discouraged.)
 wff  R 
 FrSe  A
 
Definitiondf-bnj15 27407 Define the following predicate:  R is both well-founded and set-like on  A. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( R  FrSe  A  <->  ( R  Fr  A  /\  R  Se  A ) )
 
Syntaxc-bnj18 27408 Extend class notation with the function giving: the transitive closure of  X in  A by  R. (New usage is discouraged.)
 class  trCl ( X ,  A ,  R )
 
Definitiondf-bnj18 27409* Define the function giving: the transitive closure of  X in  A by  R. This definition has been designed for facilitating verification that it is eliminable and that the $d restrictions are sound and complete. For a more readable definition see bnj882 27647. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  trCl ( X ,  A ,  R )  =  U_ f  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `  (/) )  = 
 pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) ) } U_ i  e.  dom  f ( f `  i )
 
Syntaxw-bnj19 27410 Extend wff notation with the following predicate:  B is transitive for  A and  R. (New usage is discouraged.)
 wff  TrFo
 ( B ,  A ,  R )
 
Definitiondf-bnj19 27411* Define the following predicate:  B is transitive for  A and  R. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (  TrFo ( B ,  A ,  R )  <->  A. x  e.  B  pred ( x ,  A ,  R )  C_  B )
 
16.21.1  First order logic and set theory
 
Theorembnj170 27412  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch )  <->  ( ( ps 
 /\  ch )  /\  ph )
 )
 
Theorembnj240 27413  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps  ->  ps' )   &    |-  ( ch  ->  ch' )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( ps'  /\  ch' ) )
 
Theorembnj248 27414  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( (
 ph  /\  ps )  /\  ch )  /\  th ) )
 
Theorembnj250 27415  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  (
 ( ps  /\  ch )  /\  th ) ) )
 
Theorembnj251 27416  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  ( ps  /\  ( ch  /\  th ) ) ) )
 
Theorembnj252 27417  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  ( ps  /\  ch  /\  th ) ) )
 
Theorembnj253 27418  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ph  /\ 
 ps )  /\  ch  /\ 
 th ) )
 
Theorembnj255 27419  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  ps  /\  ( ch  /\  th ) ) )
 
Theorembnj256 27420  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) )
 
Theorembnj257 27421  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  ps  /\ 
 th  /\  ch )
 )
 
Theorembnj258 27422  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ph  /\ 
 ps  /\  th )  /\  ch ) )
 
Theorembnj268 27423  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  ch  /\ 
 ps  /\  th )
 )
 
Theorembnj290 27424  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  ch  /\ 
 th  /\  ps )
 )
 
Theorembnj291 27425  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ph  /\ 
 ch  /\  th )  /\  ps ) )
 
Theorembnj312 27426  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ps  /\  ph 
 /\  ch  /\  th )
 )
 
Theorembnj334 27427  /\-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 27428  /\-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 27429  /\-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 27430  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ch 
 /\  th )  /\  ( ph  /\  ps ) ) )
 
Theorembnj446 27431  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ps 
 /\  ch  /\  th )  /\  ph ) )
 
Theorembnj21 27432* 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 27433* 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 27434 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 27435* 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 27436* 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 27437* 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 27438 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 27439 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  1o  e.  _V
 
Theorembnj115 27440 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 27441* 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 27442 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 27443 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 27444 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 27445 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 27446* 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 27447* First-order logic and set theory. Revised to remove dependence on ax-reg 7190. (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 27448 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 27449 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 27450 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 27451* 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 27452 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 27453 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 27454 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( A  i^i  { A }
 )  =  (/)
 
Theorembnj524 27455 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 27456* 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 27457* 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 27458* 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 27459 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 27460 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 27461 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 27462 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 27463 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 27464 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 27465* 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 27466  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ph )
 
Theorembnj643 27467  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ps )
 
Theorembnj645 27468  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  th )
 
Theorembnj658 27469  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ( ph  /\  ps  /\  ch ) )
 
Theorembnj667 27470  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ( ps  /\  ch  /\  th ) )
 
Theorembnj705 27471  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 ch  /\  th )  ->  ta )
 
Theorembnj706 27472  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 ch  /\  th )  ->  ta )
 
Theorembnj707 27473  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 ch  /\  th )  ->  ta )
 
Theorembnj708 27474  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( th  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 ch  /\  th )  ->  ta )
 
Theorembnj721 27475  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  ta )   =>    |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ta )
 
Theorembnj832 27476  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps ) )   &    |-  ( ph  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj833 27477  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps ) )   &    |-  ( ps  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj835 27478  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps  /\ 
 ch ) )   &    |-  ( ph  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj836 27479  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps  /\ 
 ch ) )   &    |-  ( ps  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj837 27480  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps  /\ 
 ch ) )   &    |-  ( ch  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj769 27481  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps  /\ 
 ch  /\  th )
 )   &    |-  ( ph  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj770 27482  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps  /\ 
 ch  /\  th )
 )   &    |-  ( ps  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj771 27483  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps  /\ 
 ch  /\  th )
 )   &    |-  ( ch  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj887 27484  /\-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 27485 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 27486* 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 27487 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 27488 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ( ps  <->  ph ) )  ->  ps )
 
Theorembnj927 27489 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 27490 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 27491 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 27492* 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 27493 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 27494 Technical lemma for bnj69 27729. 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 27495 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 27496  /\-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 27497 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 27498* 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 27499 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 27500* 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 ) )
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