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Theorem 3optocl 4766
Description: Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
Hypotheses
Ref Expression
3optocl.1  |-  R  =  ( D  X.  F
)
3optocl.2  |-  ( <.
x ,  y >.  =  A  ->  ( ph  <->  ps ) )
3optocl.3  |-  ( <.
z ,  w >.  =  B  ->  ( ps  <->  ch ) )
3optocl.4  |-  ( <.
v ,  u >.  =  C  ->  ( ch  <->  th ) )
3optocl.5  |-  ( ( ( x  e.  D  /\  y  e.  F
)  /\  ( z  e.  D  /\  w  e.  F )  /\  (
v  e.  D  /\  u  e.  F )
)  ->  ph )
Assertion
Ref Expression
3optocl  |-  ( ( A  e.  R  /\  B  e.  R  /\  C  e.  R )  ->  th )
Distinct variable groups:    x, y,
z, w, v, u, A    z, B, w, v, u    v, C, u    x, D, y, z, w, v, u   
x, F, y, z, w, v, u    z, R, w, v, u    ps, x, y    ch, z, w    th, v, u
Allowed substitution hints:    ph( x, y, z, w, v, u)    ps( z, w, v, u)    ch( x, y, v, u)    th( x, y, z, w)    B( x, y)    C( x, y, z, w)    R( x, y)

Proof of Theorem 3optocl
StepHypRef Expression
1 3optocl.1 . . . 4  |-  R  =  ( D  X.  F
)
2 3optocl.4 . . . . 5  |-  ( <.
v ,  u >.  =  C  ->  ( ch  <->  th ) )
32imbi2d 307 . . . 4  |-  ( <.
v ,  u >.  =  C  ->  ( (
( A  e.  R  /\  B  e.  R
)  ->  ch )  <->  ( ( A  e.  R  /\  B  e.  R
)  ->  th )
) )
4 3optocl.2 . . . . . . 7  |-  ( <.
x ,  y >.  =  A  ->  ( ph  <->  ps ) )
54imbi2d 307 . . . . . 6  |-  ( <.
x ,  y >.  =  A  ->  ( ( ( v  e.  D  /\  u  e.  F
)  ->  ph )  <->  ( (
v  e.  D  /\  u  e.  F )  ->  ps ) ) )
6 3optocl.3 . . . . . . 7  |-  ( <.
z ,  w >.  =  B  ->  ( ps  <->  ch ) )
76imbi2d 307 . . . . . 6  |-  ( <.
z ,  w >.  =  B  ->  ( (
( v  e.  D  /\  u  e.  F
)  ->  ps )  <->  ( ( v  e.  D  /\  u  e.  F
)  ->  ch )
) )
8 3optocl.5 . . . . . . 7  |-  ( ( ( x  e.  D  /\  y  e.  F
)  /\  ( z  e.  D  /\  w  e.  F )  /\  (
v  e.  D  /\  u  e.  F )
)  ->  ph )
983expia 1153 . . . . . 6  |-  ( ( ( x  e.  D  /\  y  e.  F
)  /\  ( z  e.  D  /\  w  e.  F ) )  -> 
( ( v  e.  D  /\  u  e.  F )  ->  ph )
)
101, 5, 7, 92optocl 4765 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  R )  ->  ( ( v  e.  D  /\  u  e.  F )  ->  ch ) )
1110com12 27 . . . 4  |-  ( ( v  e.  D  /\  u  e.  F )  ->  ( ( A  e.  R  /\  B  e.  R )  ->  ch ) )
121, 3, 11optocl 4764 . . 3  |-  ( C  e.  R  ->  (
( A  e.  R  /\  B  e.  R
)  ->  th )
)
1312impcom 419 . 2  |-  ( ( ( A  e.  R  /\  B  e.  R
)  /\  C  e.  R )  ->  th )
14133impa 1146 1  |-  ( ( A  e.  R  /\  B  e.  R  /\  C  e.  R )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643    X. cxp 4687
This theorem is referenced by:  ecopovtrn  6761
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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