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Theorem cbvoprab1 6144
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 1629 . . . . . 6  |-  F/ w  v  =  <. x ,  y >.
2 cbvoprab1.1 . . . . . 6  |-  F/ w ph
31, 2nfan 1846 . . . . 5  |-  F/ w
( v  =  <. x ,  y >.  /\  ph )
43nfex 1865 . . . 4  |-  F/ w E. y ( v  = 
<. x ,  y >.  /\  ph )
5 nfv 1629 . . . . . 6  |-  F/ x  v  =  <. w ,  y >.
6 cbvoprab1.2 . . . . . 6  |-  F/ x ps
75, 6nfan 1846 . . . . 5  |-  F/ x
( v  =  <. w ,  y >.  /\  ps )
87nfex 1865 . . . 4  |-  F/ x E. y ( v  = 
<. w ,  y >.  /\  ps )
9 opeq1 3984 . . . . . . 7  |-  ( x  =  w  ->  <. x ,  y >.  =  <. w ,  y >. )
109eqeq2d 2447 . . . . . 6  |-  ( x  =  w  ->  (
v  =  <. x ,  y >.  <->  v  =  <. w ,  y >.
) )
11 cbvoprab1.3 . . . . . 6  |-  ( x  =  w  ->  ( ph 
<->  ps ) )
1210, 11anbi12d 692 . . . . 5  |-  ( x  =  w  ->  (
( v  =  <. x ,  y >.  /\  ph ) 
<->  ( v  =  <. w ,  y >.  /\  ps ) ) )
1312exbidv 1636 . . . 4  |-  ( x  =  w  ->  ( E. y ( v  = 
<. x ,  y >.  /\  ph )  <->  E. y
( v  =  <. w ,  y >.  /\  ps ) ) )
144, 8, 13cbvex 1983 . . 3  |-  ( E. x E. y ( v  =  <. x ,  y >.  /\  ph ) 
<->  E. w E. y
( v  =  <. w ,  y >.  /\  ps ) )
1514opabbii 4272 . 2  |-  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }  =  { <. v ,  z >.  |  E. w E. y ( v  =  <. w ,  y
>.  /\  ps ) }
16 dfoprab2 6121 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }
17 dfoprab2 6121 . 2  |-  { <. <.
w ,  y >. ,  z >.  |  ps }  =  { <. v ,  z >.  |  E. w E. y ( v  =  <. w ,  y
>.  /\  ps ) }
1815, 16, 173eqtr4i 2466 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. w ,  y >. ,  z
>.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550   F/wnf 1553    = wceq 1652   <.cop 3817   {copab 4265   {coprab 6082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-oprab 6085
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