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Theorem cbvoprab1 5607
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 1462 . . . . . 6  |-  F/ w  v  =  <. x ,  y >.
2 cbvoprab1.1 . . . . . 6  |-  F/ w ph
31, 2nfan 1498 . . . . 5  |-  F/ w
( v  =  <. x ,  y >.  /\  ph )
43nfex 1569 . . . 4  |-  F/ w E. y ( v  = 
<. x ,  y >.  /\  ph )
5 nfv 1462 . . . . . 6  |-  F/ x  v  =  <. w ,  y >.
6 cbvoprab1.2 . . . . . 6  |-  F/ x ps
75, 6nfan 1498 . . . . 5  |-  F/ x
( v  =  <. w ,  y >.  /\  ps )
87nfex 1569 . . . 4  |-  F/ x E. y ( v  = 
<. w ,  y >.  /\  ps )
9 opeq1 3578 . . . . . . 7  |-  ( x  =  w  ->  <. x ,  y >.  =  <. w ,  y >. )
109eqeq2d 2093 . . . . . 6  |-  ( x  =  w  ->  (
v  =  <. x ,  y >.  <->  v  =  <. w ,  y >.
) )
11 cbvoprab1.3 . . . . . 6  |-  ( x  =  w  ->  ( ph 
<->  ps ) )
1210, 11anbi12d 457 . . . . 5  |-  ( x  =  w  ->  (
( v  =  <. x ,  y >.  /\  ph ) 
<->  ( v  =  <. w ,  y >.  /\  ps ) ) )
1312exbidv 1747 . . . 4  |-  ( x  =  w  ->  ( E. y ( v  = 
<. x ,  y >.  /\  ph )  <->  E. y
( v  =  <. w ,  y >.  /\  ps ) ) )
144, 8, 13cbvex 1680 . . 3  |-  ( E. x E. y ( v  =  <. x ,  y >.  /\  ph ) 
<->  E. w E. y
( v  =  <. w ,  y >.  /\  ps ) )
1514opabbii 3853 . 2  |-  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }  =  { <. v ,  z >.  |  E. w E. y ( v  =  <. w ,  y
>.  /\  ps ) }
16 dfoprab2 5583 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }
17 dfoprab2 5583 . 2  |-  { <. <.
w ,  y >. ,  z >.  |  ps }  =  { <. v ,  z >.  |  E. w E. y ( v  =  <. w ,  y
>.  /\  ps ) }
1815, 16, 173eqtr4i 2112 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. w ,  y >. ,  z
>.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   F/wnf 1390   E.wex 1422   <.cop 3409   {copab 3846   {coprab 5544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-opab 3848  df-oprab 5547
This theorem is referenced by: (None)
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