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Theorem cbvmpt2 5611
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 2194 . 2  |-  F/_ z B
2 nfcv 2194 . 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 2057 . 2  |-  ( x  =  z  ->  B  =  B )
8 cbvmpt2.5 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )
91, 2, 3, 4, 5, 6, 7, 8cbvmpt2x 5610 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 101    = wceq 1259   F/_wnfc 2181    |-> cmpt2 5542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-oprab 5544  df-mpt2 5545
This theorem is referenced by:  cbvmpt2v  5612  fmpt2co  5865
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