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Theorem optocl 4720
Description: Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)
Hypotheses
Ref Expression
optocl.1  |-  D  =  ( B  X.  C
)
optocl.2  |-  ( <.
x ,  y >.  =  A  ->  ( ph  <->  ps ) )
optocl.3  |-  ( ( x  e.  B  /\  y  e.  C )  ->  ph )
Assertion
Ref Expression
optocl  |-  ( A  e.  D  ->  ps )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y    ps, x, y
Allowed substitution hints:    ph( x, y)    D( x, y)

Proof of Theorem optocl
StepHypRef Expression
1 elxp3 4698 . . 3  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( <. x ,  y >.  =  A  /\  <. x ,  y
>.  e.  ( B  X.  C ) ) )
2 opelxp 4674 . . . . . . 7  |-  ( <.
x ,  y >.  e.  ( B  X.  C
)  <->  ( x  e.  B  /\  y  e.  C ) )
3 optocl.3 . . . . . . 7  |-  ( ( x  e.  B  /\  y  e.  C )  ->  ph )
42, 3sylbi 121 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( B  X.  C
)  ->  ph )
5 optocl.2 . . . . . 6  |-  ( <.
x ,  y >.  =  A  ->  ( ph  <->  ps ) )
64, 5imbitrid 154 . . . . 5  |-  ( <.
x ,  y >.  =  A  ->  ( <.
x ,  y >.  e.  ( B  X.  C
)  ->  ps )
)
76imp 124 . . . 4  |-  ( (
<. x ,  y >.  =  A  /\  <. x ,  y >.  e.  ( B  X.  C ) )  ->  ps )
87exlimivv 1908 . . 3  |-  ( E. x E. y (
<. x ,  y >.  =  A  /\  <. x ,  y >.  e.  ( B  X.  C ) )  ->  ps )
91, 8sylbi 121 . 2  |-  ( A  e.  ( B  X.  C )  ->  ps )
10 optocl.1 . 2  |-  D  =  ( B  X.  C
)
119, 10eleq2s 2284 1  |-  ( A  e.  D  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   E.wex 1503    e. wcel 2160   <.cop 3610    X. cxp 4642
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 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-opab 4080  df-xp 4650
This theorem is referenced by:  2optocl  4721  3optocl  4722  ecoptocl  6648  ax1rid  7906  ax0id  7907  axcnre  7910
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