ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3ecoptocl Unicode version

Theorem 3ecoptocl 6650
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 230 . . . 4  |-  ( [
<. z ,  w >. ] R  =  B  -> 
( ( A  e.  S  ->  ps )  <->  ( A  e.  S  ->  ch ) ) )
4 3ecoptocl.4 . . . . 5  |-  ( [
<. v ,  u >. ] R  =  C  -> 
( ch  <->  th )
)
54imbi2d 230 . . . 4  |-  ( [
<. v ,  u >. ] R  =  C  -> 
( ( A  e.  S  ->  ch )  <->  ( A  e.  S  ->  th ) ) )
6 3ecoptocl.2 . . . . . . 7  |-  ( [
<. x ,  y >. ] R  =  A  ->  ( ph  <->  ps )
)
76imbi2d 230 . . . . . 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 1208 . . . . . 6  |-  ( ( x  e.  D  /\  y  e.  D )  ->  ( ( ( z  e.  D  /\  w  e.  D )  /\  (
v  e.  D  /\  u  e.  D )
)  ->  ph ) )
101, 7, 9ecoptocl 6648 . . . . 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 6649 . . 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 1203 1  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   <.cop 3610    X. cxp 4642   [cec 6557   /.cqs 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-ec 6561  df-qs 6565
This theorem is referenced by:  ecovass  6670  ecoviass  6671  ecovdi  6672  ecovidi  6673  ltsonq  7427  ltanqg  7429  ltmnqg  7430  lttrsr  7791  ltsosr  7793  ltasrg  7799  mulextsr1  7810
  Copyright terms: Public domain W3C validator