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Theorem opelopab2 4301
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 462 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ch )
)
43opelopab2a 4295 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2164   <.cop 3621   {copab 4089
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-opab 4091
This theorem is referenced by:  brecop  6679  netap  7314  2oneel  7316  2omotaplemap  7317  2omotaplemst  7318  divides  11932
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