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Theorem opelopabsbOLD 4464
Description: The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
opelopabsbOLD  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] [
z  /  x ] ph )
Distinct variable groups:    x, y,
z    x, w, y
Allowed substitution hints:    ph( x, y, z, w)

Proof of Theorem opelopabsbOLD
StepHypRef Expression
1 excom 1757 . . 3  |-  ( E. x E. y (
<. z ,  w >.  = 
<. x ,  y >.  /\  ph )  <->  E. y E. x ( <. z ,  w >.  =  <. x ,  y >.  /\  ph ) )
2 vex 2960 . . . . . . 7  |-  z  e. 
_V
3 vex 2960 . . . . . . 7  |-  w  e. 
_V
42, 3opth 4436 . . . . . 6  |-  ( <.
z ,  w >.  = 
<. x ,  y >.  <->  ( z  =  x  /\  w  =  y )
)
5 equcom 1693 . . . . . . 7  |-  ( z  =  x  <->  x  =  z )
6 equcom 1693 . . . . . . 7  |-  ( w  =  y  <->  y  =  w )
75, 6anbi12ci 681 . . . . . 6  |-  ( ( z  =  x  /\  w  =  y )  <->  ( y  =  w  /\  x  =  z )
)
84, 7bitri 242 . . . . 5  |-  ( <.
z ,  w >.  = 
<. x ,  y >.  <->  ( y  =  w  /\  x  =  z )
)
98anbi1i 678 . . . 4  |-  ( (
<. z ,  w >.  = 
<. x ,  y >.  /\  ph )  <->  ( (
y  =  w  /\  x  =  z )  /\  ph ) )
1092exbii 1594 . . 3  |-  ( E. y E. x (
<. z ,  w >.  = 
<. x ,  y >.  /\  ph )  <->  E. y E. x ( ( y  =  w  /\  x  =  z )  /\  ph ) )
111, 10bitri 242 . 2  |-  ( E. x E. y (
<. z ,  w >.  = 
<. x ,  y >.  /\  ph )  <->  E. y E. x ( ( y  =  w  /\  x  =  z )  /\  ph ) )
12 elopab 4463 . 2  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  E. x E. y ( <. z ,  w >.  =  <. x ,  y >.  /\  ph ) )
13 2sb5 2189 . 2  |-  ( [ w  /  y ] [ z  /  x ] ph  <->  E. y E. x
( ( y  =  w  /\  x  =  z )  /\  ph ) )
1411, 12, 133bitr4i 270 1  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] [
z  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653   [wsb 1659    e. wcel 1726   <.cop 3818   {copab 4266
This theorem is referenced by:  brabsbOLD  4465  inopab  5006  cnvopab  5275  brabsb2  26712
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 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
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 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-opab 4268
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