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Theorem cbvopab 4087
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 1605 . . . . 5  |-  F/ z  v  =  <. x ,  y >.
2 cbvopab.1 . . . . 5  |-  F/ z
ph
31, 2nfan 1771 . . . 4  |-  F/ z ( v  =  <. x ,  y >.  /\  ph )
4 nfv 1605 . . . . 5  |-  F/ w  v  =  <. x ,  y >.
5 cbvopab.2 . . . . 5  |-  F/ w ph
64, 5nfan 1771 . . . 4  |-  F/ w
( v  =  <. x ,  y >.  /\  ph )
7 nfv 1605 . . . . 5  |-  F/ x  v  =  <. z ,  w >.
8 cbvopab.3 . . . . 5  |-  F/ x ps
97, 8nfan 1771 . . . 4  |-  F/ x
( v  =  <. z ,  w >.  /\  ps )
10 nfv 1605 . . . . 5  |-  F/ y  v  =  <. z ,  w >.
11 cbvopab.4 . . . . 5  |-  F/ y ps
1210, 11nfan 1771 . . . 4  |-  F/ y ( v  =  <. z ,  w >.  /\  ps )
13 opeq12 3798 . . . . . 6  |-  ( ( x  =  z  /\  y  =  w )  -> 
<. x ,  y >.  =  <. z ,  w >. )
1413eqeq2d 2294 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( v  =  <. x ,  y >.  <->  v  =  <. z ,  w >. ) )
15 cbvopab.5 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
1614, 15anbi12d 691 . . . 4  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( v  = 
<. x ,  y >.  /\  ph )  <->  ( v  =  <. z ,  w >.  /\  ps ) ) )
173, 6, 9, 12, 16cbvex2 1945 . . 3  |-  ( E. x E. y ( v  =  <. x ,  y >.  /\  ph ) 
<->  E. z E. w
( v  =  <. z ,  w >.  /\  ps ) )
1817abbii 2395 . 2  |-  { v  |  E. x E. y ( v  = 
<. x ,  y >.  /\  ph ) }  =  { v  |  E. z E. w ( v  =  <. z ,  w >.  /\  ps ) }
19 df-opab 4078 . 2  |-  { <. x ,  y >.  |  ph }  =  { v  |  E. x E. y
( v  =  <. x ,  y >.  /\  ph ) }
20 df-opab 4078 . 2  |-  { <. z ,  w >.  |  ps }  =  { v  |  E. z E. w
( v  =  <. z ,  w >.  /\  ps ) }
2118, 19, 203eqtr4i 2313 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  w >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528   F/wnf 1531    = wceq 1623   {cab 2269   <.cop 3643   {copab 4076
This theorem is referenced by:  cbvopabv  4088  dfrel4  23204  aomclem8  27159
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-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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