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Theorem cbvoprab3 5608
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 1437 . . . . . 6  |-  F/ w  v  =  <. x ,  y >.
2 cbvoprab3.1 . . . . . 6  |-  F/ w ph
31, 2nfan 1473 . . . . 5  |-  F/ w
( v  =  <. x ,  y >.  /\  ph )
43nfex 1544 . . . 4  |-  F/ w E. y ( v  = 
<. x ,  y >.  /\  ph )
54nfex 1544 . . 3  |-  F/ w E. x E. y ( v  =  <. x ,  y >.  /\  ph )
6 nfv 1437 . . . . . 6  |-  F/ z  v  =  <. x ,  y >.
7 cbvoprab3.2 . . . . . 6  |-  F/ z ps
86, 7nfan 1473 . . . . 5  |-  F/ z ( v  =  <. x ,  y >.  /\  ps )
98nfex 1544 . . . 4  |-  F/ z E. y ( v  =  <. x ,  y
>.  /\  ps )
109nfex 1544 . . 3  |-  F/ z E. x E. y
( v  =  <. x ,  y >.  /\  ps )
11 cbvoprab3.3 . . . . 5  |-  ( z  =  w  ->  ( ph 
<->  ps ) )
1211anbi2d 445 . . . 4  |-  ( z  =  w  ->  (
( v  =  <. x ,  y >.  /\  ph ) 
<->  ( v  =  <. x ,  y >.  /\  ps ) ) )
13122exbidv 1764 . . 3  |-  ( z  =  w  ->  ( E. x E. y ( v  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. y
( v  =  <. x ,  y >.  /\  ps ) ) )
145, 10, 13cbvopab2 3859 . 2  |-  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }  =  { <. v ,  w >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ps ) }
15 dfoprab2 5580 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. v ,  z >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ph ) }
16 dfoprab2 5580 . 2  |-  { <. <.
x ,  y >. ,  w >.  |  ps }  =  { <. v ,  w >.  |  E. x E. y ( v  =  <. x ,  y
>.  /\  ps ) }
1714, 15, 163eqtr4i 2086 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  w >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259   F/wnf 1365   E.wex 1397   <.cop 3406   {copab 3845   {coprab 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-oprab 5544
This theorem is referenced by:  cbvoprab3v  5609  tposoprab  5926  erovlem  6229
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