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Theorem cbvopab 3915
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 1467 . . . . 5  |-  F/ z  v  =  <. x ,  y >.
2 cbvopab.1 . . . . 5  |-  F/ z
ph
31, 2nfan 1503 . . . 4  |-  F/ z ( v  =  <. x ,  y >.  /\  ph )
4 nfv 1467 . . . . 5  |-  F/ w  v  =  <. x ,  y >.
5 cbvopab.2 . . . . 5  |-  F/ w ph
64, 5nfan 1503 . . . 4  |-  F/ w
( v  =  <. x ,  y >.  /\  ph )
7 nfv 1467 . . . . 5  |-  F/ x  v  =  <. z ,  w >.
8 cbvopab.3 . . . . 5  |-  F/ x ps
97, 8nfan 1503 . . . 4  |-  F/ x
( v  =  <. z ,  w >.  /\  ps )
10 nfv 1467 . . . . 5  |-  F/ y  v  =  <. z ,  w >.
11 cbvopab.4 . . . . 5  |-  F/ y ps
1210, 11nfan 1503 . . . 4  |-  F/ y ( v  =  <. z ,  w >.  /\  ps )
13 opeq12 3630 . . . . . 6  |-  ( ( x  =  z  /\  y  =  w )  -> 
<. x ,  y >.  =  <. z ,  w >. )
1413eqeq2d 2100 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( v  =  <. x ,  y >.  <->  v  =  <. z ,  w >. ) )
15 cbvopab.5 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
1614, 15anbi12d 458 . . . 4  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( v  = 
<. x ,  y >.  /\  ph )  <->  ( v  =  <. z ,  w >.  /\  ps ) ) )
173, 6, 9, 12, 16cbvex2 1846 . . 3  |-  ( E. x E. y ( v  =  <. x ,  y >.  /\  ph ) 
<->  E. z E. w
( v  =  <. z ,  w >.  /\  ps ) )
1817abbii 2204 . 2  |-  { v  |  E. x E. y ( v  = 
<. x ,  y >.  /\  ph ) }  =  { v  |  E. z E. w ( v  =  <. z ,  w >.  /\  ps ) }
19 df-opab 3906 . 2  |-  { <. x ,  y >.  |  ph }  =  { v  |  E. x E. y
( v  =  <. x ,  y >.  /\  ph ) }
20 df-opab 3906 . 2  |-  { <. z ,  w >.  |  ps }  =  { v  |  E. z E. w
( v  =  <. z ,  w >.  /\  ps ) }
2118, 19, 203eqtr4i 2119 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  w >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290   F/wnf 1395   E.wex 1427   {cab 2075   <.cop 3453   {copab 3904
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-opab 3906
This theorem is referenced by:  cbvopabv  3916  opelopabsb  4096
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