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Theorem cbvmpt2 6154
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)
Hypotheses
Ref Expression
cbvmpt2.1  |-  F/_ z C
cbvmpt2.2  |-  F/_ w C
cbvmpt2.3  |-  F/_ x D
cbvmpt2.4  |-  F/_ y D
cbvmpt2.5  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )
Assertion
Ref Expression
cbvmpt2  |-  ( 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
Allowed substitution hints:    C( x, y, z, w)    D( x, y, z, w)

Proof of Theorem cbvmpt2
StepHypRef Expression
1 nfcv 2574 . 2  |-  F/_ z B
2 nfcv 2574 . 2  |-  F/_ x B
3 cbvmpt2.1 . 2  |-  F/_ z C
4 cbvmpt2.2 . 2  |-  F/_ w C
5 cbvmpt2.3 . 2  |-  F/_ x D
6 cbvmpt2.4 . 2  |-  F/_ y D
7 eqidd 2439 . 2  |-  ( x  =  z  ->  B  =  B )
8 cbvmpt2.5 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )
91, 2, 3, 4, 5, 6, 7, 8cbvmpt2x 6153 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    /\ wa 360    = wceq 1653   F/_wnfc 2561    e. cmpt2 6086
This theorem is referenced by:  cbvmpt2v  6155  fmpt2co  6433  xpf1o  7272  cnfcomlem  7659  fseqenlem1  7910  gsumdixp  15720  evlslem4  16569  cnmpt2t  17710  cnmptk2  17723  fmucnd  18327  fsum2cn  18906  relexpsucr  25135  fmuldfeqlem1  27702
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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