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Theorem cbvopab 4360
Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
Hypotheses
Ref Expression
cbvopab.1  |-  F/ z
ph
cbvopab.2  |-  F/ w ph
cbvopab.3  |-  F/ x ps
cbvopab.4  |-  F/ y ps
cbvopab.5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
cbvopab  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  w >.  |  ps }
Distinct variable group:    x, y, z, w
Allowed substitution hints:    ph( x, y, z, w)    ps( x, y, z, w)

Proof of Theorem cbvopab
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 nfv 1673 . . . . 5  |-  F/ z  v  =  <. x ,  y >.
2 cbvopab.1 . . . . 5  |-  F/ z
ph
31, 2nfan 1861 . . . 4  |-  F/ z ( v  =  <. x ,  y >.  /\  ph )
4 nfv 1673 . . . . 5  |-  F/ w  v  =  <. x ,  y >.
5 cbvopab.2 . . . . 5  |-  F/ w ph
64, 5nfan 1861 . . . 4  |-  F/ w
( v  =  <. x ,  y >.  /\  ph )
7 nfv 1673 . . . . 5  |-  F/ x  v  =  <. z ,  w >.
8 cbvopab.3 . . . . 5  |-  F/ x ps
97, 8nfan 1861 . . . 4  |-  F/ x
( v  =  <. z ,  w >.  /\  ps )
10 nfv 1673 . . . . 5  |-  F/ y  v  =  <. z ,  w >.
11 cbvopab.4 . . . . 5  |-  F/ y ps
1210, 11nfan 1861 . . . 4  |-  F/ y ( v  =  <. z ,  w >.  /\  ps )
13 opeq12 4061 . . . . . 6  |-  ( ( x  =  z  /\  y  =  w )  -> 
<. x ,  y >.  =  <. z ,  w >. )
1413eqeq2d 2454 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( v  =  <. x ,  y >.  <->  v  =  <. z ,  w >. ) )
15 cbvopab.5 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
1614, 15anbi12d 710 . . . 4  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( v  = 
<. x ,  y >.  /\  ph )  <->  ( v  =  <. z ,  w >.  /\  ps ) ) )
173, 6, 9, 12, 16cbvex2 1976 . . 3  |-  ( E. x E. y ( v  =  <. x ,  y >.  /\  ph ) 
<->  E. z E. w
( v  =  <. z ,  w >.  /\  ps ) )
1817abbii 2555 . 2  |-  { v  |  E. x E. y ( v  = 
<. x ,  y >.  /\  ph ) }  =  { v  |  E. z E. w ( v  =  <. z ,  w >.  /\  ps ) }
19 df-opab 4351 . 2  |-  { <. x ,  y >.  |  ph }  =  { v  |  E. x E. y
( v  =  <. x ,  y >.  /\  ph ) }
20 df-opab 4351 . 2  |-  { <. z ,  w >.  |  ps }  =  { v  |  E. z E. w
( v  =  <. z ,  w >.  /\  ps ) }
2118, 19, 203eqtr4i 2473 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  w >.  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586   F/wnf 1589   {cab 2429   <.cop 3883   {copab 4349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-opab 4351
This theorem is referenced by:  cbvopabv  4361  dfrel4  25937  aomclem8  29414
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