Theorem cbvmpt2 5583

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

Step	Hyp	Ref	Expression
1		nfcv 2178	. 2 |- F/_ z B
2		nfcv 2178	. 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 2041	. 2 |- ( x = z -> B = B )
8		cbvmpt2.5	. 2 |- ( ( x = z /\ y = w ) -> C = D )
9	1, 2, 3, 4, 5, 6, 7, 8	cbvmpt2x 5582	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   F/_wnfc 2165   |-> 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:  cbvmpt2v  5584  fmpt2co  5837