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Theorem opelopab2a 4280
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 2343 . . . . 5  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
2 eleq1 2343 . . . . 5  |-  ( y  =  B  ->  (
y  e.  D  <->  B  e.  D ) )
31, 2bi2anan9 843 . . . 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 691 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( x  e.  C  /\  y  e.  D )  /\  ph ) 
<->  ( ( A  e.  C  /\  B  e.  D )  /\  ps ) ) )
65opelopabga 4278 . 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 850 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   {copab 4076
This theorem is referenced by:  opelopab2  4285  brab2a  4738  brab2ga  4763  prdsleval  13376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078
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