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Theorem cbvopab 4124
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 1610 . . . . 5  |-  F/ z  v  =  <. x ,  y >.
2 cbvopab.1 . . . . 5  |-  F/ z
ph
31, 2nfan 1800 . . . 4  |-  F/ z ( v  =  <. x ,  y >.  /\  ph )
4 nfv 1610 . . . . 5  |-  F/ w  v  =  <. x ,  y >.
5 cbvopab.2 . . . . 5  |-  F/ w ph
64, 5nfan 1800 . . . 4  |-  F/ w
( v  =  <. x ,  y >.  /\  ph )
7 nfv 1610 . . . . 5  |-  F/ x  v  =  <. z ,  w >.
8 cbvopab.3 . . . . 5  |-  F/ x ps
97, 8nfan 1800 . . . 4  |-  F/ x
( v  =  <. z ,  w >.  /\  ps )
10 nfv 1610 . . . . 5  |-  F/ y  v  =  <. z ,  w >.
11 cbvopab.4 . . . . 5  |-  F/ y ps
1210, 11nfan 1800 . . . 4  |-  F/ y ( v  =  <. z ,  w >.  /\  ps )
13 opeq12 3835 . . . . . 6  |-  ( ( x  =  z  /\  y  =  w )  -> 
<. x ,  y >.  =  <. z ,  w >. )
1413eqeq2d 2327 . . . . 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 1977 . . 3  |-  ( E. x E. y ( v  =  <. x ,  y >.  /\  ph ) 
<->  E. z E. w
( v  =  <. z ,  w >.  /\  ps ) )
1817abbii 2428 . 2  |-  { v  |  E. x E. y ( v  = 
<. x ,  y >.  /\  ph ) }  =  { v  |  E. z E. w ( v  =  <. z ,  w >.  /\  ps ) }
19 df-opab 4115 . 2  |-  { <. x ,  y >.  |  ph }  =  { v  |  E. x E. y
( v  =  <. x ,  y >.  /\  ph ) }
20 df-opab 4115 . 2  |-  { <. z ,  w >.  |  ps }  =  { v  |  E. z E. w
( v  =  <. z ,  w >.  /\  ps ) }
2118, 19, 203eqtr4i 2346 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  w >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1532   F/wnf 1535    = wceq 1633   {cab 2302   <.cop 3677   {copab 4113
This theorem is referenced by:  cbvopabv  4125  dfrel4  23188  aomclem8  26307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-opab 4115
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