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Theorem cbvoprab3 5897
Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)
Hypotheses
Ref Expression
cbvoprab3.1  |-  F/ w ph
cbvoprab3.2  |-  F/ z ps
cbvoprab3.3  |-  ( z  =  w  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvoprab3  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  w >.  |  ps }
Distinct variable groups:    x, z, w   
y, z, w
Allowed substitution hints:    ph( x, y, z, w)    ps( x, y, z, w)

Proof of Theorem cbvoprab3
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 nfv 1508 . . . . . 6  |-  F/ w  v  =  <. x ,  y >.
2 cbvoprab3.1 . . . . . 6  |-  F/ w ph
31, 2nfan 1545 . . . . 5  |-  F/ w
( v  =  <. x ,  y >.  /\  ph )
43nfex 1617 . . . 4  |-  F/ w E. y ( v  = 
<. x ,  y >.  /\  ph )
54nfex 1617 . . 3  |-  F/ w E. x E. y ( v  =  <. x ,  y >.  /\  ph )
6 nfv 1508 . . . . . 6  |-  F/ z  v  =  <. x ,  y >.
7 cbvoprab3.2 . . . . . 6  |-  F/ z ps
86, 7nfan 1545 . . . . 5  |-  F/ z ( v  =  <. x ,  y >.  /\  ps )
98nfex 1617 . . . 4  |-  F/ z E. y ( v  =  <. x ,  y
>.  /\  ps )
109nfex 1617 . . 3  |-  F/ z E. x E. y
( v  =  <. x ,  y >.  /\  ps )
11 cbvoprab3.3 . . . . 5  |-  ( z  =  w  ->  ( ph 
<->  ps ) )
1211anbi2d 460 . . . 4  |-  ( z  =  w  ->  (
( v  =  <. x ,  y >.  /\  ph ) 
<->  ( v  =  <. x ,  y >.  /\  ps ) ) )
13122exbidv 1848 . . 3  |-  ( z  =  w  ->  ( E. x E. y ( v  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. y
( v  =  <. x ,  y >.  /\  ps ) ) )
145, 10, 13cbvopab2 4038 . 2  |-  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }  =  { <. v ,  w >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ps ) }
15 dfoprab2 5868 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }
16 dfoprab2 5868 . 2  |-  { <. <.
x ,  y >. ,  w >.  |  ps }  =  { <. v ,  w >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ps ) }
1714, 15, 163eqtr4i 2188 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  w >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335   F/wnf 1440   E.wex 1472   <.cop 3563   {copab 4024   {coprab 5825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-opab 4026  df-oprab 5828
This theorem is referenced by:  cbvoprab3v  5898  tposoprab  6227  erovlem  6572
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