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Theorem cbvoprab12 5927
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 1521 . . . . 5  |-  F/ w  u  =  <. x ,  y >.
2 cbvoprab12.1 . . . . 5  |-  F/ w ph
31, 2nfan 1558 . . . 4  |-  F/ w
( u  =  <. x ,  y >.  /\  ph )
4 nfv 1521 . . . . 5  |-  F/ v  u  =  <. x ,  y >.
5 cbvoprab12.2 . . . . 5  |-  F/ v
ph
64, 5nfan 1558 . . . 4  |-  F/ v ( u  =  <. x ,  y >.  /\  ph )
7 nfv 1521 . . . . 5  |-  F/ x  u  =  <. w ,  v >.
8 cbvoprab12.3 . . . . 5  |-  F/ x ps
97, 8nfan 1558 . . . 4  |-  F/ x
( u  =  <. w ,  v >.  /\  ps )
10 nfv 1521 . . . . 5  |-  F/ y  u  =  <. w ,  v >.
11 cbvoprab12.4 . . . . 5  |-  F/ y ps
1210, 11nfan 1558 . . . 4  |-  F/ y ( u  =  <. w ,  v >.  /\  ps )
13 opeq12 3767 . . . . . 6  |-  ( ( x  =  w  /\  y  =  v )  -> 
<. x ,  y >.  =  <. w ,  v
>. )
1413eqeq2d 2182 . . . . 5  |-  ( ( x  =  w  /\  y  =  v )  ->  ( u  =  <. x ,  y >.  <->  u  =  <. w ,  v >.
) )
15 cbvoprab12.5 . . . . 5  |-  ( ( x  =  w  /\  y  =  v )  ->  ( ph  <->  ps )
)
1614, 15anbi12d 470 . . . 4  |-  ( ( x  =  w  /\  y  =  v )  ->  ( ( u  = 
<. x ,  y >.  /\  ph )  <->  ( u  =  <. w ,  v
>.  /\  ps ) ) )
173, 6, 9, 12, 16cbvex2 1915 . . 3  |-  ( E. x E. y ( u  =  <. x ,  y >.  /\  ph ) 
<->  E. w E. v
( u  =  <. w ,  v >.  /\  ps ) )
1817opabbii 4056 . 2  |-  { <. u ,  z >.  |  E. x E. y ( u  =  <. x ,  y
>.  /\  ph ) }  =  { <. u ,  z >.  |  E. w E. v ( u  =  <. w ,  v
>.  /\  ps ) }
19 dfoprab2 5900 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. u ,  z >.  |  E. x E. y ( u  =  <. x ,  y
>.  /\  ph ) }
20 dfoprab2 5900 . 2  |-  { <. <.
w ,  v >. ,  z >.  |  ps }  =  { <. u ,  z >.  |  E. w E. v ( u  =  <. w ,  v
>.  /\  ps ) }
2118, 19, 203eqtr4i 2201 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. w ,  v >. ,  z
>.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348   F/wnf 1453   E.wex 1485   <.cop 3586   {copab 4049   {coprab 5854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-oprab 5857
This theorem is referenced by:  cbvoprab12v  5928  cbvmpox  5931  dfoprab4f  6172  fmpox  6179  tposoprab  6259
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