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Theorem 3optocl 4895
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 308 . . . 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 308 . . . . . 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 308 . . . . . 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 1155 . . . . . 6  |-  ( ( ( x  e.  D  /\  y  e.  F
)  /\  ( z  e.  D  /\  w  e.  F ) )  -> 
( ( v  e.  D  /\  u  e.  F )  ->  ph )
)
101, 5, 7, 92optocl 4894 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  R )  ->  ( ( v  e.  D  /\  u  e.  F )  ->  ch ) )
1110com12 29 . . . 4  |-  ( ( v  e.  D  /\  u  e.  F )  ->  ( ( A  e.  R  /\  B  e.  R )  ->  ch ) )
121, 3, 11optocl 4893 . . 3  |-  ( C  e.  R  ->  (
( A  e.  R  /\  B  e.  R
)  ->  th )
)
1312impcom 420 . 2  |-  ( ( ( A  e.  R  /\  B  e.  R
)  /\  C  e.  R )  ->  th )
14133impa 1148 1  |-  ( ( A  e.  R  /\  B  e.  R  /\  C  e.  R )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   <.cop 3761    X. cxp 4817
This theorem is referenced by:  ecopovtrn  6944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-opab 4209  df-xp 4825
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