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Theorem cbvmpt2 6371
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 2629 . 2  |-  F/_ z B
2 nfcv 2629 . 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 2468 . 2  |-  ( x  =  z  ->  B  =  B )
8 cbvmpt2.5 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )
91, 2, 3, 4, 5, 6, 7, 8cbvmpt2x 6370 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  A ,  w  e.  B  |->  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   F/_wnfc 2615    |-> cmpt2 6297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-opab 4512  df-oprab 6299  df-mpt2 6300
This theorem is referenced by:  cbvmpt2v  6372  fnmpt2ovd  6873  fmpt2co  6878  mpt2curryd  7010  fvmpt2curryd  7012  xpf1o  7691  cnfcomlem  8155  cnfcomlemOLD  8163  fseqenlem1  8417  gsumdixpOLD  17129  gsumdixp  17130  evlslem4OLD  18043  evlslem4  18044  madugsum  19014  cnmpt2t  20042  cnmptk2  20055  fmucnd  20663  fsum2cn  21243  relexpsucr  28869  fmuldfeqlem1  31446
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