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Theorem opelopab2a 4471
Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypothesis
Ref Expression
opelopabga.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
opelopab2a  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <->  ps ) )
Distinct variable groups:    x, y, A    x, B, y    ps, x, y    x, C, y   
x, D, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem opelopab2a
StepHypRef Expression
1 eleq1 2497 . . . . 5  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
2 eleq1 2497 . . . . 5  |-  ( y  =  B  ->  (
y  e.  D  <->  B  e.  D ) )
31, 2bi2anan9 845 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  e.  C  /\  y  e.  D )  <->  ( A  e.  C  /\  B  e.  D ) ) )
4 opelopabga.1 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
53, 4anbi12d 693 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( x  e.  C  /\  y  e.  D )  /\  ph ) 
<->  ( ( A  e.  C  /\  B  e.  D )  /\  ps ) ) )
65opelopabga 4469 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <-> 
( ( A  e.  C  /\  B  e.  D )  /\  ps ) ) )
76bianabs 852 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   <.cop 3818   {copab 4266
This theorem is referenced by:  opelopab2  4476  brab2a  4928  brab2ga  4952  prdsleval  13700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-opab 4268
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