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Theorem cbvoprab1 5811
Description: Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)
Hypotheses
Ref Expression
cbvoprab1.1  |-  F/ w ph
cbvoprab1.2  |-  F/ x ps
cbvoprab1.3  |-  ( x  =  w  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvoprab1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. w ,  y >. ,  z
>.  |  ps }
Distinct variable group:    x, y, z, w
Allowed substitution hints:    ph( x, y, z, w)    ps( x, y, z, w)

Proof of Theorem cbvoprab1
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 nfv 1493 . . . . . 6  |-  F/ w  v  =  <. x ,  y >.
2 cbvoprab1.1 . . . . . 6  |-  F/ w ph
31, 2nfan 1529 . . . . 5  |-  F/ w
( v  =  <. x ,  y >.  /\  ph )
43nfex 1601 . . . 4  |-  F/ w E. y ( v  = 
<. x ,  y >.  /\  ph )
5 nfv 1493 . . . . . 6  |-  F/ x  v  =  <. w ,  y >.
6 cbvoprab1.2 . . . . . 6  |-  F/ x ps
75, 6nfan 1529 . . . . 5  |-  F/ x
( v  =  <. w ,  y >.  /\  ps )
87nfex 1601 . . . 4  |-  F/ x E. y ( v  = 
<. w ,  y >.  /\  ps )
9 opeq1 3675 . . . . . . 7  |-  ( x  =  w  ->  <. x ,  y >.  =  <. w ,  y >. )
109eqeq2d 2129 . . . . . 6  |-  ( x  =  w  ->  (
v  =  <. x ,  y >.  <->  v  =  <. w ,  y >.
) )
11 cbvoprab1.3 . . . . . 6  |-  ( x  =  w  ->  ( ph 
<->  ps ) )
1210, 11anbi12d 464 . . . . 5  |-  ( x  =  w  ->  (
( v  =  <. x ,  y >.  /\  ph ) 
<->  ( v  =  <. w ,  y >.  /\  ps ) ) )
1312exbidv 1781 . . . 4  |-  ( x  =  w  ->  ( E. y ( v  = 
<. x ,  y >.  /\  ph )  <->  E. y
( v  =  <. w ,  y >.  /\  ps ) ) )
144, 8, 13cbvex 1714 . . 3  |-  ( E. x E. y ( v  =  <. x ,  y >.  /\  ph ) 
<->  E. w E. y
( v  =  <. w ,  y >.  /\  ps ) )
1514opabbii 3965 . 2  |-  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }  =  { <. v ,  z >.  |  E. w E. y ( v  =  <. w ,  y
>.  /\  ps ) }
16 dfoprab2 5786 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }
17 dfoprab2 5786 . 2  |-  { <. <.
w ,  y >. ,  z >.  |  ps }  =  { <. v ,  z >.  |  E. w E. y ( v  =  <. w ,  y
>.  /\  ps ) }
1815, 16, 173eqtr4i 2148 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. w ,  y >. ,  z
>.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   F/wnf 1421   E.wex 1453   <.cop 3500   {copab 3958   {coprab 5743
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960  df-oprab 5746
This theorem is referenced by: (None)
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