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Theorem opelopab2a 4716
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 2517 . . . . 5  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
2 eleq1 2517 . . . . 5  |-  ( y  =  B  ->  (
y  e.  D  <->  B  e.  D ) )
31, 2bi2anan9 884 . . . 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 717 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( x  e.  C  /\  y  e.  D )  /\  ph ) 
<->  ( ( A  e.  C  /\  B  e.  D )  /\  ps ) ) )
65opelopabga 4714 . 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 891 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   <.cop 3974   {copab 4460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-opab 4462
This theorem is referenced by:  opelopab2  4722  brab2a  4884  brab2ga  4910  prdsleval  15375  isperp  24757
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