Theorem cbvmpt2v 5584

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 3851, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)

Hypotheses

Ref	Expression
cbvmpt2v.1	|- ( x = z -> C = E )
cbvmpt2v.2	|- ( y = w -> E = D )

Assertion

Ref	Expression
cbvmpt2v	|- ( 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 cbvmpt2v

Step	Hyp	Ref	Expression
1		nfcv 2178	. 2 |- F/_ z C
2		nfcv 2178	. 2 |- F/_ w C
3		nfcv 2178	. 2 |- F/_ x D
4		nfcv 2178	. 2 |- F/_ y D
5		cbvmpt2v.1	. . 3 |- ( x = z -> C = E )
6		cbvmpt2v.2	. . 3 |- ( y = w -> E = D )
7	5, 6	sylan9eq 2092	. 2 |- ( ( x = z /\ y = w ) -> C = D )
8	1, 2, 3, 4, 7	cbvmpt2 5583	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Syntax hints:   -> wi 4   = wceq 1243   |-> cmpt2 5514

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-oprab 5516  df-mpt2 5517

This theorem is referenced by:  frec2uzrdg  9195  frecuzrdgsuc  9201  resqrexlemfp1  9607  resqrex  9624