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Theorem cbvmpt2v 6155
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4302, 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 2574 . 2  |-  F/_ z C
2 nfcv 2574 . 2  |-  F/_ w C
3 nfcv 2574 . 2  |-  F/_ x D
4 nfcv 2574 . 2  |-  F/_ y D
5 cbvmpt2v.1 . . 3  |-  ( x  =  z  ->  C  =  E )
6 cbvmpt2v.2 . . 3  |-  ( y  =  w  ->  E  =  D )
75, 6sylan9eq 2490 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )
81, 2, 3, 4, 7cbvmpt2 6154 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 1653    e. cmpt2 6086
This theorem is referenced by:  seqomlem0  6709  dffi3  7439  cantnfsuc  7628  fin23lem33  8230  om2uzrdg  11301  uzrdgsuci  11305  sadcp1  12972  smupp1  12997  imasvscafn  13767  sylow1  15242  sylow2b  15262  sylow3lem5  15270  sylow3  15272  efgmval  15349  efgtf  15359  txbas  17604  bcth  19287  opnmbl  19499  mbfimaopn  19551  mbfi1fseq  19616  opsqrlem3  23650  dya2iocival  24628  sxbrsigalem5  24643  sxbrsigalem6  24644  cvmliftlem15  24990  cvmlift2  25008  opnmbllem0  26254  mblfinlem1  26255  mblfinlem2  26256  sdc  26462  tendoplcbv  31646  dvhvaddcbv  31961  dvhvscacbv  31970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4270  df-oprab 6088  df-mpt2 6089
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