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Theorem cbvoprab3 6140
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 1629 . . . . . 6  |-  F/ w  v  =  <. x ,  y >.
2 cbvoprab3.1 . . . . . 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 )
54nfex 1865 . . 3  |-  F/ w E. x E. y ( v  =  <. x ,  y >.  /\  ph )
6 nfv 1629 . . . . . 6  |-  F/ z  v  =  <. x ,  y >.
7 cbvoprab3.2 . . . . . 6  |-  F/ z ps
86, 7nfan 1846 . . . . 5  |-  F/ z ( v  =  <. x ,  y >.  /\  ps )
98nfex 1865 . . . 4  |-  F/ z E. y ( v  =  <. x ,  y
>.  /\  ps )
109nfex 1865 . . 3  |-  F/ z E. x E. y
( v  =  <. x ,  y >.  /\  ps )
11 cbvoprab3.3 . . . . 5  |-  ( z  =  w  ->  ( ph 
<->  ps ) )
1211anbi2d 685 . . . 4  |-  ( z  =  w  ->  (
( v  =  <. x ,  y >.  /\  ph ) 
<->  ( v  =  <. x ,  y >.  /\  ps ) ) )
13122exbidv 1638 . . 3  |-  ( z  =  w  ->  ( E. x E. y ( v  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. y
( v  =  <. x ,  y >.  /\  ps ) ) )
145, 10, 13cbvopab2 4271 . 2  |-  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }  =  { <. v ,  w >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ps ) }
15 dfoprab2 6113 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }
16 dfoprab2 6113 . 2  |-  { <. <.
x ,  y >. ,  w >.  |  ps }  =  { <. v ,  w >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ps ) }
1714, 15, 163eqtr4i 2465 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  w >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550   F/wnf 1553    = wceq 1652   <.cop 3809   {copab 4257   {coprab 6074
This theorem is referenced by:  cbvoprab3v  6141  tposoprab  6507  erovlem  6992
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259  df-oprab 6077
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