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Theorem opelopab2a 4264
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 2240 . . . . 5  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
2 eleq1 2240 . . . . 5  |-  ( y  =  B  ->  (
y  e.  D  <->  B  e.  D ) )
31, 2bi2anan9 606 . . . 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 473 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( x  e.  C  /\  y  e.  D )  /\  ph ) 
<->  ( ( A  e.  C  /\  B  e.  D )  /\  ps ) ) )
65opelopabga 4262 . 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 611 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    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   <.cop 3595   {copab 4062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-opab 4064
This theorem is referenced by:  opelopab2  4269  brab2a  4678  brab2ga  4700  ltdfpr  7501  aprval  13262
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