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Theorem cbvmpov 5854
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4026, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
cbvmpov.1  |-  ( x  =  z  ->  C  =  E )
cbvmpov.2  |-  ( y  =  w  ->  E  =  D )
Assertion
Ref Expression
cbvmpov  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  A ,  w  e.  B  |->  D )
Distinct variable groups:    x, w, y, z, A    w, B, x, y, z    w, C, z    x, D, y
Allowed substitution hints:    C( x, y)    D( z, w)    E( x, y, z, w)

Proof of Theorem cbvmpov
StepHypRef Expression
1 nfcv 2281 . 2  |-  F/_ z C
2 nfcv 2281 . 2  |-  F/_ w C
3 nfcv 2281 . 2  |-  F/_ x D
4 nfcv 2281 . 2  |-  F/_ y D
5 cbvmpov.1 . . 3  |-  ( x  =  z  ->  C  =  E )
6 cbvmpov.2 . . 3  |-  ( y  =  w  ->  E  =  D )
75, 6sylan9eq 2192 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )
81, 2, 3, 4, 7cbvmpo 5853 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  A ,  w  e.  B  |->  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. cmpo 5779
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3993  df-oprab 5781  df-mpo 5782
This theorem is referenced by:  frec2uzrdg  10206  frecuzrdgsuc  10211  iseqvalcbv  10254  resqrexlemfp1  10805  resqrex  10822  sqne2sq  11878  ennnfonelemnn0  11958  txbas  12453  xmetxp  12702
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