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Theorem opelopabsbOLD 4273
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.)
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 1786 . . 3  |-  ( E. x E. y (
<. z ,  w >.  = 
<. x ,  y >.  /\  ph )  <->  E. y E. x ( <. z ,  w >.  =  <. x ,  y >.  /\  ph ) )
2 vex 2791 . . . . . . 7  |-  z  e. 
_V
3 vex 2791 . . . . . . 7  |-  w  e. 
_V
42, 3opth 4245 . . . . . 6  |-  ( <.
z ,  w >.  = 
<. x ,  y >.  <->  ( z  =  x  /\  w  =  y )
)
5 equcom 1647 . . . . . . 7  |-  ( z  =  x  <->  x  =  z )
6 equcom 1647 . . . . . . 7  |-  ( w  =  y  <->  y  =  w )
75, 6anbi12ci 679 . . . . . 6  |-  ( ( z  =  x  /\  w  =  y )  <->  ( y  =  w  /\  x  =  z )
)
84, 7bitri 240 . . . . 5  |-  ( <.
z ,  w >.  = 
<. x ,  y >.  <->  ( y  =  w  /\  x  =  z )
)
98anbi1i 676 . . . 4  |-  ( (
<. z ,  w >.  = 
<. x ,  y >.  /\  ph )  <->  ( (
y  =  w  /\  x  =  z )  /\  ph ) )
1092exbii 1570 . . 3  |-  ( E. y E. x (
<. z ,  w >.  = 
<. x ,  y >.  /\  ph )  <->  E. y E. x ( ( y  =  w  /\  x  =  z )  /\  ph ) )
111, 10bitri 240 . 2  |-  ( E. x E. y (
<. z ,  w >.  = 
<. x ,  y >.  /\  ph )  <->  E. y E. x ( ( y  =  w  /\  x  =  z )  /\  ph ) )
12 elopab 4272 . 2  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  E. x E. y ( <. z ,  w >.  =  <. x ,  y >.  /\  ph ) )
13 2sb5 2051 . 2  |-  ( [ w  /  y ] [ z  /  x ] ph  <->  E. y E. x
( ( y  =  w  /\  x  =  z )  /\  ph ) )
1411, 12, 133bitr4i 268 1  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] [
z  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623   [wsb 1629    e. wcel 1684   <.cop 3643   {copab 4076
This theorem is referenced by:  brabsbOLD  4274  inopab  4816  cnvopab  5083  brabsb2  26730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078
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