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Theorem cbvmpt2 6051
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 2502 . 2  |-  F/_ z B
2 nfcv 2502 . 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 2367 . 2  |-  ( x  =  z  ->  B  =  B )
8 cbvmpt2.5 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )
91, 2, 3, 4, 5, 6, 7, 8cbvmpt2x 6050 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 1647   F/_wnfc 2489    e. cmpt2 5983
This theorem is referenced by:  cbvmpt2v  6052  fmpt2co  6330  xpf1o  7166  cnfcomlem  7549  fseqenlem1  7798  gsumdixp  15602  evlslem4  16455  cnmpt2t  17584  cnmptk2  17597  fsum2cn  18589  fmucnd  23785  relexpsucr  24613  fmuldfeqlem1  27218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-opab 4180  df-oprab 5985  df-mpt2 5986
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