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Theorem 3optocl 4587
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 229 . . . 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 229 . . . . . 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 229 . . . . . 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 1168 . . . . . 6  |-  ( ( ( x  e.  D  /\  y  e.  F
)  /\  ( z  e.  D  /\  w  e.  F ) )  -> 
( ( v  e.  D  /\  u  e.  F )  ->  ph )
)
101, 5, 7, 92optocl 4586 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  R )  ->  ( ( v  e.  D  /\  u  e.  F )  ->  ch ) )
1110com12 30 . . . 4  |-  ( ( v  e.  D  /\  u  e.  F )  ->  ( ( A  e.  R  /\  B  e.  R )  ->  ch ) )
121, 3, 11optocl 4585 . . 3  |-  ( C  e.  R  ->  (
( A  e.  R  /\  B  e.  R
)  ->  th )
)
1312impcom 124 . 2  |-  ( ( ( A  e.  R  /\  B  e.  R
)  /\  C  e.  R )  ->  th )
14133impa 1161 1  |-  ( ( A  e.  R  /\  B  e.  R  /\  C  e.  R )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   <.cop 3500    X. cxp 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960  df-xp 4515
This theorem is referenced by:  ecopovtrn  6494  ecopovtrng  6497
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