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Theorem cbvoprab2 6077
Description: Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
cbvoprab2.1  |-  F/ w ph
cbvoprab2.2  |-  F/ y ps
cbvoprab2.3  |-  ( y  =  w  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvoprab2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  w >. ,  z >.  |  ps }
Distinct variable group:    x, w, y, z
Allowed substitution hints:    ph( x, y, z, w)    ps( x, y, z, w)

Proof of Theorem cbvoprab2
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 nfv 1626 . . . . . . 7  |-  F/ w  v  =  <. <. x ,  y >. ,  z
>.
2 cbvoprab2.1 . . . . . . 7  |-  F/ w ph
31, 2nfan 1836 . . . . . 6  |-  F/ w
( v  =  <. <.
x ,  y >. ,  z >.  /\  ph )
43nfex 1855 . . . . 5  |-  F/ w E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )
5 nfv 1626 . . . . . . 7  |-  F/ y  v  =  <. <. x ,  w >. ,  z >.
6 cbvoprab2.2 . . . . . . 7  |-  F/ y ps
75, 6nfan 1836 . . . . . 6  |-  F/ y ( v  =  <. <.
x ,  w >. ,  z >.  /\  ps )
87nfex 1855 . . . . 5  |-  F/ y E. z ( v  =  <. <. x ,  w >. ,  z >.  /\  ps )
9 opeq2 3920 . . . . . . . . 9  |-  ( y  =  w  ->  <. x ,  y >.  =  <. x ,  w >. )
109opeq1d 3925 . . . . . . . 8  |-  ( y  =  w  ->  <. <. x ,  y >. ,  z
>.  =  <. <. x ,  w >. ,  z >.
)
1110eqeq2d 2391 . . . . . . 7  |-  ( y  =  w  ->  (
v  =  <. <. x ,  y >. ,  z
>. 
<->  v  =  <. <. x ,  w >. ,  z >.
) )
12 cbvoprab2.3 . . . . . . 7  |-  ( y  =  w  ->  ( ph 
<->  ps ) )
1311, 12anbi12d 692 . . . . . 6  |-  ( y  =  w  ->  (
( v  =  <. <.
x ,  y >. ,  z >.  /\  ph ) 
<->  ( v  =  <. <.
x ,  w >. ,  z >.  /\  ps )
) )
1413exbidv 1633 . . . . 5  |-  ( y  =  w  ->  ( E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. z
( v  =  <. <.
x ,  w >. ,  z >.  /\  ps )
) )
154, 8, 14cbvex 2011 . . . 4  |-  ( E. y E. z ( v  =  <. <. x ,  y >. ,  z
>.  /\  ph )  <->  E. w E. z ( v  = 
<. <. x ,  w >. ,  z >.  /\  ps ) )
1615exbii 1589 . . 3  |-  ( E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. x E. w E. z ( v  =  <. <. x ,  w >. ,  z >.  /\  ps ) )
1716abbii 2492 . 2  |-  { v  |  E. x E. y E. z ( v  =  <. <. x ,  y
>. ,  z >.  /\ 
ph ) }  =  { v  |  E. x E. w E. z
( v  =  <. <.
x ,  w >. ,  z >.  /\  ps ) }
18 df-oprab 6017 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { v  |  E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
19 df-oprab 6017 . 2  |-  { <. <.
x ,  w >. ,  z >.  |  ps }  =  { v  |  E. x E. w E. z ( v  = 
<. <. x ,  w >. ,  z >.  /\  ps ) }
2017, 18, 193eqtr4i 2410 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  w >. ,  z >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547   F/wnf 1550    = wceq 1649   {cab 2366   <.cop 3753   {coprab 6014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-oprab 6017
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