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Theorem optocl 4954
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 4930 . . 3  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( <. x ,  y >.  =  A  /\  <. x ,  y
>.  e.  ( B  X.  C ) ) )
2 opelxp 4910 . . . . . . 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 189 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( B  X.  C
)  ->  ph )
5 optocl.2 . . . . . 6  |-  ( <.
x ,  y >.  =  A  ->  ( ph  <->  ps ) )
64, 5syl5ib 212 . . . . 5  |-  ( <.
x ,  y >.  =  A  ->  ( <.
x ,  y >.  e.  ( B  X.  C
)  ->  ps )
)
76imp 420 . . . 4  |-  ( (
<. x ,  y >.  =  A  /\  <. x ,  y >.  e.  ( B  X.  C ) )  ->  ps )
87exlimivv 1646 . . 3  |-  ( E. x E. y (
<. x ,  y >.  =  A  /\  <. x ,  y >.  e.  ( B  X.  C ) )  ->  ps )
91, 8sylbi 189 . 2  |-  ( A  e.  ( B  X.  C )  ->  ps )
10 optocl.1 . 2  |-  D  =  ( B  X.  C
)
119, 10eleq2s 2530 1  |-  ( A  e.  D  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   <.cop 3819    X. cxp 4878
This theorem is referenced by:  2optocl  4955  3optocl  4956  ecoptocl  6996  ax1rid  9038  axcnre  9041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-xp 4886
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