Theorem cbvmpt2 5727

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 2228	. 2 |- F/_ z B
2		nfcv 2228	. 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 2089	. 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 5726	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   F/_wnfc 2215   |-> cmpt2 5654

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-oprab 5656  df-mpt2 5657

This theorem is referenced by:  cbvmpt2v  5728  fmpt2co  5981  xpf1o  6560