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Theorem cbvoprab1 5918
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 1605 . . . . . 6  |-  F/ w  v  =  <. x ,  y >.
2 cbvoprab1.1 . . . . . 6  |-  F/ w ph
31, 2nfan 1771 . . . . 5  |-  F/ w
( v  =  <. x ,  y >.  /\  ph )
43nfex 1767 . . . 4  |-  F/ w E. y ( v  = 
<. x ,  y >.  /\  ph )
5 nfv 1605 . . . . . 6  |-  F/ x  v  =  <. w ,  y >.
6 cbvoprab1.2 . . . . . 6  |-  F/ x ps
75, 6nfan 1771 . . . . 5  |-  F/ x
( v  =  <. w ,  y >.  /\  ps )
87nfex 1767 . . . 4  |-  F/ x E. y ( v  = 
<. w ,  y >.  /\  ps )
9 opeq1 3796 . . . . . . 7  |-  ( x  =  w  ->  <. x ,  y >.  =  <. w ,  y >. )
109eqeq2d 2294 . . . . . 6  |-  ( x  =  w  ->  (
v  =  <. x ,  y >.  <->  v  =  <. w ,  y >.
) )
11 cbvoprab1.3 . . . . . 6  |-  ( x  =  w  ->  ( ph 
<->  ps ) )
1210, 11anbi12d 691 . . . . 5  |-  ( x  =  w  ->  (
( v  =  <. x ,  y >.  /\  ph ) 
<->  ( v  =  <. w ,  y >.  /\  ps ) ) )
1312exbidv 1612 . . . 4  |-  ( x  =  w  ->  ( E. y ( v  = 
<. x ,  y >.  /\  ph )  <->  E. y
( v  =  <. w ,  y >.  /\  ps ) ) )
144, 8, 13cbvex 1925 . . 3  |-  ( E. x E. y ( v  =  <. x ,  y >.  /\  ph ) 
<->  E. w E. y
( v  =  <. w ,  y >.  /\  ps ) )
1514opabbii 4083 . 2  |-  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }  =  { <. v ,  z >.  |  E. w E. y ( v  =  <. w ,  y
>.  /\  ps ) }
16 dfoprab2 5895 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }
17 dfoprab2 5895 . 2  |-  { <. <.
w ,  y >. ,  z >.  |  ps }  =  { <. v ,  z >.  |  E. w E. y ( v  =  <. w ,  y
>.  /\  ps ) }
1815, 16, 173eqtr4i 2313 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. w ,  y >. ,  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 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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