Theorem cbvmpt2 5603

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 2219	. 2 |- F/_ z B
2		nfcv 2219	. 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 2082	. 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 5602	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   F/_wnfc 2206   |-> cmpt2 5534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840  df-oprab 5536  df-mpt2 5537

This theorem is referenced by:  cbvmpt2v  5604  fmpt2co  5857