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Theorem 2ecoptocl 6996
Description: Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.)
Hypotheses
Ref Expression
2ecoptocl.1  |-  S  =  ( ( C  X.  D ) /. R
)
2ecoptocl.2  |-  ( [
<. x ,  y >. ] R  =  A  ->  ( ph  <->  ps )
)
2ecoptocl.3  |-  ( [
<. z ,  w >. ] R  =  B  -> 
( ps  <->  ch )
)
2ecoptocl.4  |-  ( ( ( x  e.  C  /\  y  e.  D
)  /\  ( z  e.  C  /\  w  e.  D ) )  ->  ph )
Assertion
Ref Expression
2ecoptocl  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ch )
Distinct variable groups:    x, y,
z, w, A    z, B, w    x, C, y, z, w    x, D, y, z, w    z, S, w    x, R, y, z, w    ps, x, y    ch, z, w
Allowed substitution hints:    ph( x, y, z, w)    ps( z, w)    ch( x, y)    B( x, y)    S( x, y)

Proof of Theorem 2ecoptocl
StepHypRef Expression
1 2ecoptocl.1 . . 3  |-  S  =  ( ( C  X.  D ) /. R
)
2 2ecoptocl.3 . . . 4  |-  ( [
<. z ,  w >. ] R  =  B  -> 
( ps  <->  ch )
)
32imbi2d 309 . . 3  |-  ( [
<. z ,  w >. ] R  =  B  -> 
( ( A  e.  S  ->  ps )  <->  ( A  e.  S  ->  ch ) ) )
4 2ecoptocl.2 . . . . . 6  |-  ( [
<. x ,  y >. ] R  =  A  ->  ( ph  <->  ps )
)
54imbi2d 309 . . . . 5  |-  ( [
<. x ,  y >. ] R  =  A  ->  ( ( ( z  e.  C  /\  w  e.  D )  ->  ph )  <->  ( ( z  e.  C  /\  w  e.  D
)  ->  ps )
) )
6 2ecoptocl.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, 7ecoptocl 6995 . . . 4  |-  ( A  e.  S  ->  (
( z  e.  C  /\  w  e.  D
)  ->  ps )
)
98com12 30 . . 3  |-  ( ( z  e.  C  /\  w  e.  D )  ->  ( A  e.  S  ->  ps ) )
101, 3, 9ecoptocl 6995 . 2  |-  ( B  e.  S  ->  ( A  e.  S  ->  ch ) )
1110impcom 421 1  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   <.cop 3818    X. cxp 4877   [cec 6904   /.cqs 6905
This theorem is referenced by:  3ecoptocl  6997  ecovcom  7016  addclsr  8959  mulclsr  8960  ltsosr  8970  mulgt0sr  8981
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 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
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 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-cnv 4887  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-ec 6908  df-qs 6912
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