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Theorem cbvoprab12 6366
Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypotheses
Ref Expression
cbvoprab12.1  |-  F/ w ph
cbvoprab12.2  |-  F/ v
ph
cbvoprab12.3  |-  F/ x ps
cbvoprab12.4  |-  F/ y ps
cbvoprab12.5  |-  ( ( x  =  w  /\  y  =  v )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
cbvoprab12  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. w ,  v >. ,  z
>.  |  ps }
Distinct variable group:    x, y, z, w, v
Allowed substitution hints:    ph( x, y, z, w, v)    ps( x, y, z, w, v)

Proof of Theorem cbvoprab12
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 nfv 1683 . . . . 5  |-  F/ w  u  =  <. x ,  y >.
2 cbvoprab12.1 . . . . 5  |-  F/ w ph
31, 2nfan 1875 . . . 4  |-  F/ w
( u  =  <. x ,  y >.  /\  ph )
4 nfv 1683 . . . . 5  |-  F/ v  u  =  <. x ,  y >.
5 cbvoprab12.2 . . . . 5  |-  F/ v
ph
64, 5nfan 1875 . . . 4  |-  F/ v ( u  =  <. x ,  y >.  /\  ph )
7 nfv 1683 . . . . 5  |-  F/ x  u  =  <. w ,  v >.
8 cbvoprab12.3 . . . . 5  |-  F/ x ps
97, 8nfan 1875 . . . 4  |-  F/ x
( u  =  <. w ,  v >.  /\  ps )
10 nfv 1683 . . . . 5  |-  F/ y  u  =  <. w ,  v >.
11 cbvoprab12.4 . . . . 5  |-  F/ y ps
1210, 11nfan 1875 . . . 4  |-  F/ y ( u  =  <. w ,  v >.  /\  ps )
13 opeq12 4221 . . . . . 6  |-  ( ( x  =  w  /\  y  =  v )  -> 
<. x ,  y >.  =  <. w ,  v
>. )
1413eqeq2d 2481 . . . . 5  |-  ( ( x  =  w  /\  y  =  v )  ->  ( u  =  <. x ,  y >.  <->  u  =  <. w ,  v >.
) )
15 cbvoprab12.5 . . . . 5  |-  ( ( x  =  w  /\  y  =  v )  ->  ( ph  <->  ps )
)
1614, 15anbi12d 710 . . . 4  |-  ( ( x  =  w  /\  y  =  v )  ->  ( ( u  = 
<. x ,  y >.  /\  ph )  <->  ( u  =  <. w ,  v
>.  /\  ps ) ) )
173, 6, 9, 12, 16cbvex2 2001 . . 3  |-  ( E. x E. y ( u  =  <. x ,  y >.  /\  ph ) 
<->  E. w E. v
( u  =  <. w ,  v >.  /\  ps ) )
1817opabbii 4517 . 2  |-  { <. u ,  z >.  |  E. x E. y ( u  =  <. x ,  y
>.  /\  ph ) }  =  { <. u ,  z >.  |  E. w E. v ( u  =  <. w ,  v
>.  /\  ps ) }
19 dfoprab2 6338 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. u ,  z >.  |  E. x E. y ( u  =  <. x ,  y
>.  /\  ph ) }
20 dfoprab2 6338 . 2  |-  { <. <.
w ,  v >. ,  z >.  |  ps }  =  { <. u ,  z >.  |  E. w E. v ( u  =  <. w ,  v
>.  /\  ps ) }
2118, 19, 203eqtr4i 2506 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. w ,  v >. ,  z
>.  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596   F/wnf 1599   <.cop 4039   {copab 4510   {coprab 6296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-opab 4512  df-oprab 6299
This theorem is referenced by:  cbvoprab12v  6367  cbvmpt2x  6370  dfoprab4f  6853  fmpt2x  6861  tposoprab  7003  cbvmpt2x2  32404
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