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Theorem optocl 4764
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 4739 . . 3  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( <. x ,  y >.  =  A  /\  <. x ,  y
>.  e.  ( B  X.  C ) ) )
2 opelxp 4719 . . . . . . 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 187 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( B  X.  C
)  ->  ph )
5 optocl.2 . . . . . 6  |-  ( <.
x ,  y >.  =  A  ->  ( ph  <->  ps ) )
64, 5syl5ib 210 . . . . 5  |-  ( <.
x ,  y >.  =  A  ->  ( <.
x ,  y >.  e.  ( B  X.  C
)  ->  ps )
)
76imp 418 . . . 4  |-  ( (
<. x ,  y >.  =  A  /\  <. x ,  y >.  e.  ( B  X.  C ) )  ->  ps )
87exlimivv 1667 . . 3  |-  ( E. x E. y (
<. x ,  y >.  =  A  /\  <. x ,  y >.  e.  ( B  X.  C ) )  ->  ps )
91, 8sylbi 187 . 2  |-  ( A  e.  ( B  X.  C )  ->  ps )
10 optocl.1 . 2  |-  D  =  ( B  X.  C
)
119, 10eleq2s 2375 1  |-  ( A  e.  D  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   <.cop 3643    X. cxp 4687
This theorem is referenced by:  2optocl  4765  3optocl  4766  ecoptocl  6748  ax1rid  8783  axcnre  8786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695
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