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Theorem 2ecoptocl 6954
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 308 . . 3  |-  ( [
<. z ,  w >. ] R  =  B  -> 
( ( A  e.  S  ->  ps )  <->  ( A  e.  S  ->  ch ) ) )
4 2ecoptocl.2 . . . . . 6  |-  ( [
<. x ,  y >. ] R  =  A  ->  ( ph  <->  ps )
)
54imbi2d 308 . . . . 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 424 . . . . 5  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( ( z  e.  C  /\  w  e.  D )  ->  ph )
)
81, 5, 7ecoptocl 6953 . . . 4  |-  ( A  e.  S  ->  (
( z  e.  C  /\  w  e.  D
)  ->  ps )
)
98com12 29 . . 3  |-  ( ( z  e.  C  /\  w  e.  D )  ->  ( A  e.  S  ->  ps ) )
101, 3, 9ecoptocl 6953 . 2  |-  ( B  e.  S  ->  ( A  e.  S  ->  ch ) )
1110impcom 420 1  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   <.cop 3777    X. cxp 4835   [cec 6862   /.cqs 6863
This theorem is referenced by:  3ecoptocl  6955  ecovcom  6974  addclsr  8914  mulclsr  8915  ltsosr  8925  mulgt0sr  8936
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ec 6866  df-qs 6870
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