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Theorem cbvmpt2 5925
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 2419 . 2  |-  F/_ z B
2 nfcv 2419 . 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 2284 . 2  |-  ( x  =  z  ->  B  =  B )
8 cbvmpt2.5 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )
91, 2, 3, 4, 5, 6, 7, 8cbvmpt2x 5924 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 358    = wceq 1623   F/_wnfc 2406    e. cmpt2 5860
This theorem is referenced by:  cbvmpt2v  5926  fmpt2co  6202  xpf1o  7023  cnfcomlem  7402  fseqenlem1  7651  gsumdixp  15392  evlslem4  16245  cnmpt2t  17367  cnmptk2  17380  fsum2cn  18375  relexpsucr  24026  fmuldfeqlem1  27712
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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