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Theorem cbvmpo 5930
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)
Hypotheses
Ref Expression
cbvmpo.1  |-  F/_ z C
cbvmpo.2  |-  F/_ w C
cbvmpo.3  |-  F/_ x D
cbvmpo.4  |-  F/_ y D
cbvmpo.5  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )
Assertion
Ref Expression
cbvmpo  |-  ( 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 cbvmpo
StepHypRef Expression
1 nfcv 2312 . 2  |-  F/_ z B
2 nfcv 2312 . 2  |-  F/_ x B
3 cbvmpo.1 . 2  |-  F/_ z C
4 cbvmpo.2 . 2  |-  F/_ w C
5 cbvmpo.3 . 2  |-  F/_ x D
6 cbvmpo.4 . 2  |-  F/_ y D
7 eqidd 2171 . 2  |-  ( x  =  z  ->  B  =  B )
8 cbvmpo.5 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )
91, 2, 3, 4, 5, 6, 7, 8cbvmpox 5929 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 103    = wceq 1348   F/_wnfc 2299    e. cmpo 5853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-opab 4049  df-oprab 5855  df-mpo 5856
This theorem is referenced by:  cbvmpov  5931  fnmpoovd  6192  fmpoco  6193  xpf1o  6819  cnmpt2t  13048
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