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Theorem cbvoprab2 5608
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 1462 . . . . . . 7  |-  F/ w  v  =  <. <. x ,  y >. ,  z
>.
2 cbvoprab2.1 . . . . . . 7  |-  F/ w ph
31, 2nfan 1498 . . . . . 6  |-  F/ w
( v  =  <. <.
x ,  y >. ,  z >.  /\  ph )
43nfex 1569 . . . . 5  |-  F/ w E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )
5 nfv 1462 . . . . . . 7  |-  F/ y  v  =  <. <. x ,  w >. ,  z >.
6 cbvoprab2.2 . . . . . . 7  |-  F/ y ps
75, 6nfan 1498 . . . . . 6  |-  F/ y ( v  =  <. <.
x ,  w >. ,  z >.  /\  ps )
87nfex 1569 . . . . 5  |-  F/ y E. z ( v  =  <. <. x ,  w >. ,  z >.  /\  ps )
9 opeq2 3579 . . . . . . . . 9  |-  ( y  =  w  ->  <. x ,  y >.  =  <. x ,  w >. )
109opeq1d 3584 . . . . . . . 8  |-  ( y  =  w  ->  <. <. x ,  y >. ,  z
>.  =  <. <. x ,  w >. ,  z >.
)
1110eqeq2d 2093 . . . . . . 7  |-  ( y  =  w  ->  (
v  =  <. <. x ,  y >. ,  z
>. 
<->  v  =  <. <. x ,  w >. ,  z >.
) )
12 cbvoprab2.3 . . . . . . 7  |-  ( y  =  w  ->  ( ph 
<->  ps ) )
1311, 12anbi12d 457 . . . . . 6  |-  ( y  =  w  ->  (
( v  =  <. <.
x ,  y >. ,  z >.  /\  ph ) 
<->  ( v  =  <. <.
x ,  w >. ,  z >.  /\  ps )
) )
1413exbidv 1747 . . . . 5  |-  ( y  =  w  ->  ( E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. z
( v  =  <. <.
x ,  w >. ,  z >.  /\  ps )
) )
154, 8, 14cbvex 1680 . . . 4  |-  ( E. y E. z ( v  =  <. <. x ,  y >. ,  z
>.  /\  ph )  <->  E. w E. z ( v  = 
<. <. x ,  w >. ,  z >.  /\  ps ) )
1615exbii 1537 . . 3  |-  ( E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. x E. w E. z ( v  =  <. <. x ,  w >. ,  z >.  /\  ps ) )
1716abbii 2195 . 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 5547 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { v  |  E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
19 df-oprab 5547 . 2  |-  { <. <.
x ,  w >. ,  z >.  |  ps }  =  { v  |  E. x E. w E. z ( v  = 
<. <. x ,  w >. ,  z >.  /\  ps ) }
2017, 18, 193eqtr4i 2112 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  w >. ,  z >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   F/wnf 1390   E.wex 1422   {cab 2068   <.cop 3409   {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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
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-sn 3412  df-pr 3413  df-op 3415  df-oprab 5547
This theorem is referenced by: (None)
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