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Theorem cbvmpt2v 5728
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 3933, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
cbvmpt2v.1  |-  ( x  =  z  ->  C  =  E )
cbvmpt2v.2  |-  ( y  =  w  ->  E  =  D )
Assertion
Ref Expression
cbvmpt2v  |-  ( 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 cbvmpt2v
StepHypRef Expression
1 nfcv 2228 . 2  |-  F/_ z C
2 nfcv 2228 . 2  |-  F/_ w C
3 nfcv 2228 . 2  |-  F/_ x D
4 nfcv 2228 . 2  |-  F/_ y D
5 cbvmpt2v.1 . . 3  |-  ( x  =  z  ->  C  =  E )
6 cbvmpt2v.2 . . 3  |-  ( y  =  w  ->  E  =  D )
75, 6sylan9eq 2140 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )
81, 2, 3, 4, 7cbvmpt2 5727 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 1289    |-> cmpt2 5654
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-oprab 5656  df-mpt2 5657
This theorem is referenced by:  frec2uzrdg  9816  frecuzrdgsuc  9821  iseqvalcbv  9872  resqrexlemfp1  10442  resqrex  10459  sqne2sq  11433
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