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Theorem cbvoprab12 5920
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 1605 . . . . 5  |-  F/ w  u  =  <. x ,  y >.
2 cbvoprab12.1 . . . . 5  |-  F/ w ph
31, 2nfan 1771 . . . 4  |-  F/ w
( u  =  <. x ,  y >.  /\  ph )
4 nfv 1605 . . . . 5  |-  F/ v  u  =  <. x ,  y >.
5 cbvoprab12.2 . . . . 5  |-  F/ v
ph
64, 5nfan 1771 . . . 4  |-  F/ v ( u  =  <. x ,  y >.  /\  ph )
7 nfv 1605 . . . . 5  |-  F/ x  u  =  <. w ,  v >.
8 cbvoprab12.3 . . . . 5  |-  F/ x ps
97, 8nfan 1771 . . . 4  |-  F/ x
( u  =  <. w ,  v >.  /\  ps )
10 nfv 1605 . . . . 5  |-  F/ y  u  =  <. w ,  v >.
11 cbvoprab12.4 . . . . 5  |-  F/ y ps
1210, 11nfan 1771 . . . 4  |-  F/ y ( u  =  <. w ,  v >.  /\  ps )
13 opeq12 3798 . . . . . 6  |-  ( ( x  =  w  /\  y  =  v )  -> 
<. x ,  y >.  =  <. w ,  v
>. )
1413eqeq2d 2294 . . . . 5  |-  ( ( x  =  w  /\  y  =  v )  ->  ( u  =  <. x ,  y >.  <->  u  =  <. w ,  v >.
) )
15 cbvoprab12.5 . . . . 5  |-  ( ( x  =  w  /\  y  =  v )  ->  ( ph  <->  ps )
)
1614, 15anbi12d 691 . . . 4  |-  ( ( x  =  w  /\  y  =  v )  ->  ( ( u  = 
<. x ,  y >.  /\  ph )  <->  ( u  =  <. w ,  v
>.  /\  ps ) ) )
173, 6, 9, 12, 16cbvex2 1945 . . 3  |-  ( E. x E. y ( u  =  <. x ,  y >.  /\  ph ) 
<->  E. w E. v
( u  =  <. w ,  v >.  /\  ps ) )
1817opabbii 4083 . 2  |-  { <. u ,  z >.  |  E. x E. y ( u  =  <. x ,  y
>.  /\  ph ) }  =  { <. u ,  z >.  |  E. w E. v ( u  =  <. w ,  v
>.  /\  ps ) }
19 dfoprab2 5895 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. u ,  z >.  |  E. x E. y ( u  =  <. x ,  y
>.  /\  ph ) }
20 dfoprab2 5895 . 2  |-  { <. <.
w ,  v >. ,  z >.  |  ps }  =  { <. u ,  z >.  |  E. w E. v ( u  =  <. w ,  v
>.  /\  ps ) }
2118, 19, 203eqtr4i 2313 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. w ,  v >. ,  z
>.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528   F/wnf 1531    = wceq 1623   <.cop 3643   {copab 4076   {coprab 5859
This theorem is referenced by:  cbvoprab12v  5921  cbvmpt2x  5924  dfoprab4f  6178  fmpt2x  6190  tposoprab  6270
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-ne 2448  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  df-oprab 5862
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