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Theorem cbvoprab2 5919
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 1605 . . . . . . 7  |-  F/ w  v  =  <. <. x ,  y >. ,  z
>.
2 cbvoprab2.1 . . . . . . 7  |-  F/ w ph
31, 2nfan 1771 . . . . . 6  |-  F/ w
( v  =  <. <.
x ,  y >. ,  z >.  /\  ph )
43nfex 1767 . . . . 5  |-  F/ w E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )
5 nfv 1605 . . . . . . 7  |-  F/ y  v  =  <. <. x ,  w >. ,  z >.
6 cbvoprab2.2 . . . . . . 7  |-  F/ y ps
75, 6nfan 1771 . . . . . 6  |-  F/ y ( v  =  <. <.
x ,  w >. ,  z >.  /\  ps )
87nfex 1767 . . . . 5  |-  F/ y E. z ( v  =  <. <. x ,  w >. ,  z >.  /\  ps )
9 opeq2 3797 . . . . . . . . 9  |-  ( y  =  w  ->  <. x ,  y >.  =  <. x ,  w >. )
109opeq1d 3802 . . . . . . . 8  |-  ( y  =  w  ->  <. <. x ,  y >. ,  z
>.  =  <. <. x ,  w >. ,  z >.
)
1110eqeq2d 2294 . . . . . . 7  |-  ( y  =  w  ->  (
v  =  <. <. x ,  y >. ,  z
>. 
<->  v  =  <. <. x ,  w >. ,  z >.
) )
12 cbvoprab2.3 . . . . . . 7  |-  ( y  =  w  ->  ( ph 
<->  ps ) )
1311, 12anbi12d 691 . . . . . 6  |-  ( y  =  w  ->  (
( v  =  <. <.
x ,  y >. ,  z >.  /\  ph ) 
<->  ( v  =  <. <.
x ,  w >. ,  z >.  /\  ps )
) )
1413exbidv 1612 . . . . 5  |-  ( y  =  w  ->  ( E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. z
( v  =  <. <.
x ,  w >. ,  z >.  /\  ps )
) )
154, 8, 14cbvex 1925 . . . 4  |-  ( E. y E. z ( v  =  <. <. x ,  y >. ,  z
>.  /\  ph )  <->  E. w E. z ( v  = 
<. <. x ,  w >. ,  z >.  /\  ps ) )
1615exbii 1569 . . 3  |-  ( E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. x E. w E. z ( v  =  <. <. x ,  w >. ,  z >.  /\  ps ) )
1716abbii 2395 . 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 5862 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { v  |  E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
19 df-oprab 5862 . 2  |-  { <. <.
x ,  w >. ,  z >.  |  ps }  =  { v  |  E. x E. w E. z ( v  = 
<. <. x ,  w >. ,  z >.  /\  ps ) }
2017, 18, 193eqtr4i 2313 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  w >. ,  z >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528   F/wnf 1531    = wceq 1623   {cab 2269   <.cop 3643   {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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-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-oprab 5862
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