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Theorem 2optocl 4515
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 228 . . 3  |-  ( <.
z ,  w >.  =  B  ->  ( ( A  e.  R  ->  ps )  <->  ( A  e.  R  ->  ch )
) )
4 2optocl.2 . . . . . 6  |-  ( <.
x ,  y >.  =  A  ->  ( ph  <->  ps ) )
54imbi2d 228 . . . . 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 113 . . . . 5  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( ( z  e.  C  /\  w  e.  D )  ->  ph )
)
81, 5, 7optocl 4514 . . . 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 4514 . 2  |-  ( B  e.  R  ->  ( A  e.  R  ->  ch ) )
1110impcom 123 1  |-  ( ( A  e.  R  /\  B  e.  R )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   <.cop 3449    X. cxp 4436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-xp 4444
This theorem is referenced by:  3optocl  4516  ecopovsym  6388  ecopovsymg  6391  th3qlem2  6395  axaddcom  7405
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