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Theorem 2optocl 4955
Description: Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
Hypotheses
Ref Expression
2optocl.1  |-  R  =  ( C  X.  D
)
2optocl.2  |-  ( <.
x ,  y >.  =  A  ->  ( ph  <->  ps ) )
2optocl.3  |-  ( <.
z ,  w >.  =  B  ->  ( ps  <->  ch ) )
2optocl.4  |-  ( ( ( x  e.  C  /\  y  e.  D
)  /\  ( z  e.  C  /\  w  e.  D ) )  ->  ph )
Assertion
Ref Expression
2optocl  |-  ( ( A  e.  R  /\  B  e.  R )  ->  ch )
Distinct variable groups:    x, y,
z, w, A    z, B, w    x, C, y, z, w    x, D, y, z, w    ps, x, y    ch, z, w   
z, R, w
Allowed substitution hints:    ph( x, y, z, w)    ps( z, w)    ch( x, y)    B( x, y)    R( x, y)

Proof of Theorem 2optocl
StepHypRef Expression
1 2optocl.1 . . 3  |-  R  =  ( C  X.  D
)
2 2optocl.3 . . . 4  |-  ( <.
z ,  w >.  =  B  ->  ( ps  <->  ch ) )
32imbi2d 309 . . 3  |-  ( <.
z ,  w >.  =  B  ->  ( ( A  e.  R  ->  ps )  <->  ( A  e.  R  ->  ch )
) )
4 2optocl.2 . . . . . 6  |-  ( <.
x ,  y >.  =  A  ->  ( ph  <->  ps ) )
54imbi2d 309 . . . . 5  |-  ( <.
x ,  y >.  =  A  ->  ( ( ( z  e.  C  /\  w  e.  D
)  ->  ph )  <->  ( (
z  e.  C  /\  w  e.  D )  ->  ps ) ) )
6 2optocl.4 . . . . . 6  |-  ( ( ( x  e.  C  /\  y  e.  D
)  /\  ( z  e.  C  /\  w  e.  D ) )  ->  ph )
76ex 425 . . . . 5  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( ( z  e.  C  /\  w  e.  D )  ->  ph )
)
81, 5, 7optocl 4954 . . . 4  |-  ( A  e.  R  ->  (
( z  e.  C  /\  w  e.  D
)  ->  ps )
)
98com12 30 . . 3  |-  ( ( z  e.  C  /\  w  e.  D )  ->  ( A  e.  R  ->  ps ) )
101, 3, 9optocl 4954 . 2  |-  ( B  e.  R  ->  ( A  e.  R  ->  ch ) )
1110impcom 421 1  |-  ( ( A  e.  R  /\  B  e.  R )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   <.cop 3819    X. cxp 4878
This theorem is referenced by:  3optocl  4956  ecopovsym  7008  th3qlem2  7013  axaddf  9022  axmulf  9023
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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