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Theorem opelopab2 4707
Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopab2.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
opelopab2.2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
opelopab2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <->  ch ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, D, y    ch, x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem opelopab2
StepHypRef Expression
1 opelopab2.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
2 opelopab2.2 . . 3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
31, 2sylan9bb 699 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ch )
)
43opelopab2a 4702 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3981   {copab 4447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-opab 4449
This theorem is referenced by:  brecop  7293  divides  13639  cmtvalN  33162  cvrval  33220
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