Theorem cbvmpov 6096

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4182, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)

Hypotheses

Ref	Expression
cbvmpov.1	|- ( x = z -> C = E )
cbvmpov.2	|- ( y = w -> E = D )

Assertion

Ref	Expression
cbvmpov	|- ( 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   w, C, z   x, D, y

Allowed substitution hints:   C( x, y)   D( z, w)   E( x, y, z, w)

Proof of Theorem cbvmpov

Step	Hyp	Ref	Expression
1		nfcv 2372	. 2 |- F/_ z C
2		nfcv 2372	. 2 |- F/_ w C
3		nfcv 2372	. 2 |- F/_ x D
4		nfcv 2372	. 2 |- F/_ y D
5		cbvmpov.1	. . 3 |- ( x = z -> C = E )
6		cbvmpov.2	. . 3 |- ( y = w -> E = D )
7	5, 6	sylan9eq 2282	. 2 |- ( ( x = z /\ y = w ) -> C = D )
8	1, 2, 3, 4, 7	cbvmpo 6095	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Syntax hints:   -> wi 4   = wceq 1395   e. cmpo 6015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-opab 4149  df-oprab 6017  df-mpo 6018

This theorem is referenced by:  frec2uzrdg  10661  frecuzrdgsuc  10666  iseqvalcbv  10711  resqrexlemfp1  11560  resqrex  11577  sqne2sq  12739  ennnfonelemnn0  13033  nninfdc  13064  txbas  14972  xmetxp  15221  mpomulcn  15280