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Theorem opelopabsbALT 4042
Description: The law of concretion in terms of substitutions. Less general than opelopabsb 4043, but having a much shorter proof. (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
opelopabsbALT  |-  ( <.
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 opelopabsbALT
StepHypRef Expression
1 excom 1595 . . 3  |-  ( E. x E. y (
<. z ,  w >.  = 
<. x ,  y >.  /\  ph )  <->  E. y E. x ( <. z ,  w >.  =  <. x ,  y >.  /\  ph ) )
2 vex 2613 . . . . . . 7  |-  z  e. 
_V
3 vex 2613 . . . . . . 7  |-  w  e. 
_V
42, 3opth 4020 . . . . . 6  |-  ( <.
z ,  w >.  = 
<. x ,  y >.  <->  ( z  =  x  /\  w  =  y )
)
5 equcom 1635 . . . . . . 7  |-  ( z  =  x  <->  x  =  z )
6 equcom 1635 . . . . . . 7  |-  ( w  =  y  <->  y  =  w )
75, 6anbi12ci 449 . . . . . 6  |-  ( ( z  =  x  /\  w  =  y )  <->  ( y  =  w  /\  x  =  z )
)
84, 7bitri 182 . . . . 5  |-  ( <.
z ,  w >.  = 
<. x ,  y >.  <->  ( y  =  w  /\  x  =  z )
)
98anbi1i 446 . . . 4  |-  ( (
<. z ,  w >.  = 
<. x ,  y >.  /\  ph )  <->  ( (
y  =  w  /\  x  =  z )  /\  ph ) )
1092exbii 1538 . . 3  |-  ( E. y E. x (
<. z ,  w >.  = 
<. x ,  y >.  /\  ph )  <->  E. y E. x ( ( y  =  w  /\  x  =  z )  /\  ph ) )
111, 10bitri 182 . 2  |-  ( E. x E. y (
<. z ,  w >.  = 
<. x ,  y >.  /\  ph )  <->  E. y E. x ( ( y  =  w  /\  x  =  z )  /\  ph ) )
12 elopab 4041 . 2  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  E. x E. y ( <. z ,  w >.  =  <. x ,  y >.  /\  ph ) )
13 2sb5 1902 . 2  |-  ( [ w  /  y ] [ z  /  x ] ph  <->  E. y E. x
( ( y  =  w  /\  x  =  z )  /\  ph ) )
1411, 12, 133bitr4i 210 1  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] [
z  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   [wsb 1687   <.cop 3419   {copab 3858
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-opab 3860
This theorem is referenced by:  inopab  4516  cnvopab  4776
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