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Theorem 3ecoptocl 6998
Description: Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.)
Hypotheses
Ref Expression
3ecoptocl.1  |-  S  =  ( ( D  X.  D ) /. R
)
3ecoptocl.2  |-  ( [
<. x ,  y >. ] R  =  A  ->  ( ph  <->  ps )
)
3ecoptocl.3  |-  ( [
<. z ,  w >. ] R  =  B  -> 
( ps  <->  ch )
)
3ecoptocl.4  |-  ( [
<. v ,  u >. ] R  =  C  -> 
( ch  <->  th )
)
3ecoptocl.5  |-  ( ( ( x  e.  D  /\  y  e.  D
)  /\  ( z  e.  D  /\  w  e.  D )  /\  (
v  e.  D  /\  u  e.  D )
)  ->  ph )
Assertion
Ref Expression
3ecoptocl  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  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   
z, S, w, v, u    x, R, y, z, 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)    S( x, y)

Proof of Theorem 3ecoptocl
StepHypRef Expression
1 3ecoptocl.1 . . . 4  |-  S  =  ( ( D  X.  D ) /. R
)
2 3ecoptocl.3 . . . . 5  |-  ( [
<. z ,  w >. ] R  =  B  -> 
( ps  <->  ch )
)
32imbi2d 309 . . . 4  |-  ( [
<. z ,  w >. ] R  =  B  -> 
( ( A  e.  S  ->  ps )  <->  ( A  e.  S  ->  ch ) ) )
4 3ecoptocl.4 . . . . 5  |-  ( [
<. v ,  u >. ] R  =  C  -> 
( ch  <->  th )
)
54imbi2d 309 . . . 4  |-  ( [
<. v ,  u >. ] R  =  C  -> 
( ( A  e.  S  ->  ch )  <->  ( A  e.  S  ->  th ) ) )
6 3ecoptocl.2 . . . . . . 7  |-  ( [
<. x ,  y >. ] R  =  A  ->  ( ph  <->  ps )
)
76imbi2d 309 . . . . . 6  |-  ( [
<. x ,  y >. ] R  =  A  ->  ( ( ( ( z  e.  D  /\  w  e.  D )  /\  ( v  e.  D  /\  u  e.  D
) )  ->  ph )  <->  ( ( ( z  e.  D  /\  w  e.  D )  /\  (
v  e.  D  /\  u  e.  D )
)  ->  ps )
) )
8 3ecoptocl.5 . . . . . . 7  |-  ( ( ( x  e.  D  /\  y  e.  D
)  /\  ( z  e.  D  /\  w  e.  D )  /\  (
v  e.  D  /\  u  e.  D )
)  ->  ph )
983expib 1157 . . . . . 6  |-  ( ( x  e.  D  /\  y  e.  D )  ->  ( ( ( z  e.  D  /\  w  e.  D )  /\  (
v  e.  D  /\  u  e.  D )
)  ->  ph ) )
101, 7, 9ecoptocl 6996 . . . . 5  |-  ( A  e.  S  ->  (
( ( z  e.  D  /\  w  e.  D )  /\  (
v  e.  D  /\  u  e.  D )
)  ->  ps )
)
1110com12 30 . . . 4  |-  ( ( ( z  e.  D  /\  w  e.  D
)  /\  ( v  e.  D  /\  u  e.  D ) )  -> 
( A  e.  S  ->  ps ) )
121, 3, 5, 112ecoptocl 6997 . . 3  |-  ( ( B  e.  S  /\  C  e.  S )  ->  ( A  e.  S  ->  th ) )
1312com12 30 . 2  |-  ( A  e.  S  ->  (
( B  e.  S  /\  C  e.  S
)  ->  th )
)
14133impib 1152 1  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   <.cop 3819    X. cxp 4878   [cec 6905   /.cqs 6906
This theorem is referenced by:  ecovass  7018  ecovdi  7019  ltsosr  8971  ltasr  8977
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 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
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 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-ec 6909  df-qs 6913
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