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Theorem cbvmpt2v 5926
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4110, 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 2419 . 2  |-  F/_ z C
2 nfcv 2419 . 2  |-  F/_ w C
3 nfcv 2419 . 2  |-  F/_ x D
4 nfcv 2419 . 2  |-  F/_ y D
5 cbvmpt2v.1 . . 3  |-  ( x  =  z  ->  C  =  E )
6 cbvmpt2v.2 . . 3  |-  ( y  =  w  ->  E  =  D )
75, 6sylan9eq 2335 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )
81, 2, 3, 4, 7cbvmpt2 5925 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 1623    e. cmpt2 5860
This theorem is referenced by:  seqomlem0  6461  dffi3  7184  cantnfsuc  7371  fin23lem33  7971  om2uzrdg  11019  uzrdgsuci  11023  sadcp1  12646  smupp1  12671  imasvscafn  13439  sylow1  14914  sylow2b  14934  sylow3lem5  14942  sylow3  14944  efgmval  15021  efgtf  15031  txbas  17262  bcth  18751  opnmbl  18957  mbfimaopn  19011  mbfi1fseq  19076  opsqrlem3  22722  dya2iocival  23576  cvmliftlem15  23829  cvmlift2  23847  sdc  26454  tendoplcbv  30964  dvhvaddcbv  31279  dvhvscacbv  31288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-oprab 5862  df-mpt2 5863
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