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Theorem 3ecoptocl 6379
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 228 . . . 4  |-  ( [
<. z ,  w >. ] R  =  B  -> 
( ( A  e.  S  ->  ps )  <->  ( A  e.  S  ->  ch ) ) )
4 3ecoptocl.4 . . . . 5  |-  ( [
<. v ,  u >. ] R  =  C  -> 
( ch  <->  th )
)
54imbi2d 228 . . . 4  |-  ( [
<. v ,  u >. ] R  =  C  -> 
( ( A  e.  S  ->  ch )  <->  ( A  e.  S  ->  th ) ) )
6 3ecoptocl.2 . . . . . . 7  |-  ( [
<. x ,  y >. ] R  =  A  ->  ( ph  <->  ps )
)
76imbi2d 228 . . . . . 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 1146 . . . . . 6  |-  ( ( x  e.  D  /\  y  e.  D )  ->  ( ( ( z  e.  D  /\  w  e.  D )  /\  (
v  e.  D  /\  u  e.  D )
)  ->  ph ) )
101, 7, 9ecoptocl 6377 . . . . 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 6378 . . 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 1141 1  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   <.cop 3449    X. cxp 4436   [cec 6288   /.cqs 6289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-ec 6292  df-qs 6296
This theorem is referenced by:  ecovass  6399  ecoviass  6400  ecovdi  6401  ecovidi  6402  ltsonq  6955  ltanqg  6957  ltmnqg  6958  lttrsr  7306  ltsosr  7308  ltasrg  7314  mulextsr1  7324
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