MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3ecoptocl Unicode version

Theorem 3ecoptocl 6934
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 308 . . . 4  |-  ( [
<. z ,  w >. ] R  =  B  -> 
( ( A  e.  S  ->  ps )  <->  ( A  e.  S  ->  ch ) ) )
4 3ecoptocl.4 . . . . 5  |-  ( [
<. v ,  u >. ] R  =  C  -> 
( ch  <->  th )
)
54imbi2d 308 . . . 4  |-  ( [
<. v ,  u >. ] R  =  C  -> 
( ( A  e.  S  ->  ch )  <->  ( A  e.  S  ->  th ) ) )
6 3ecoptocl.2 . . . . . . 7  |-  ( [
<. x ,  y >. ] R  =  A  ->  ( ph  <->  ps )
)
76imbi2d 308 . . . . . 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 1156 . . . . . 6  |-  ( ( x  e.  D  /\  y  e.  D )  ->  ( ( ( z  e.  D  /\  w  e.  D )  /\  (
v  e.  D  /\  u  e.  D )
)  ->  ph ) )
101, 7, 9ecoptocl 6932 . . . . 5  |-  ( A  e.  S  ->  (
( ( z  e.  D  /\  w  e.  D )  /\  (
v  e.  D  /\  u  e.  D )
)  ->  ps )
)
1110com12 29 . . . 4  |-  ( ( ( z  e.  D  /\  w  e.  D
)  /\  ( v  e.  D  /\  u  e.  D ) )  -> 
( A  e.  S  ->  ps ) )
121, 3, 5, 112ecoptocl 6933 . . 3  |-  ( ( B  e.  S  /\  C  e.  S )  ->  ( A  e.  S  ->  th ) )
1312com12 29 . 2  |-  ( A  e.  S  ->  (
( B  e.  S  /\  C  e.  S
)  ->  th )
)
14133impib 1151 1  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   <.cop 3762    X. cxp 4818   [cec 6841   /.cqs 6842
This theorem is referenced by:  ecovass  6954  ecovdi  6955  ltsosr  8904  ltasr  8910
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
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 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-xp 4826  df-cnv 4828  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-ec 6845  df-qs 6849
  Copyright terms: Public domain W3C validator