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Theorem 2ecoptocl 6749
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 307 . . 3  |-  ( [
<. z ,  w >. ] R  =  B  -> 
( ( A  e.  S  ->  ps )  <->  ( A  e.  S  ->  ch ) ) )
4 2ecoptocl.2 . . . . . 6  |-  ( [
<. x ,  y >. ] R  =  A  ->  ( ph  <->  ps )
)
54imbi2d 307 . . . . 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 423 . . . . 5  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( ( z  e.  C  /\  w  e.  D )  ->  ph )
)
81, 5, 7ecoptocl 6748 . . . 4  |-  ( A  e.  S  ->  (
( z  e.  C  /\  w  e.  D
)  ->  ps )
)
98com12 27 . . 3  |-  ( ( z  e.  C  /\  w  e.  D )  ->  ( A  e.  S  ->  ps ) )
101, 3, 9ecoptocl 6748 . 2  |-  ( B  e.  S  ->  ( A  e.  S  ->  ch ) )
1110impcom 419 1  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643    X. cxp 4687   [cec 6658   /.cqs 6659
This theorem is referenced by:  3ecoptocl  6750  ecovcom  6769  addclsr  8705  mulclsr  8706  ltsosr  8716  mulgt0sr  8727
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-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-ec 6662  df-qs 6666
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