Theorem mpteq12dva 3861

Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)

Hypotheses

Ref	Expression
mpteq12dv.1	|- ( ph -> A = C )
mpteq12dva.2	|- ( ( ph /\ x e. A ) -> B = D )

Assertion

Ref	Expression
mpteq12dva	|- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Distinct variable group:   ph, x

Allowed substitution hints:   A( x)   B( x)   C( x)   D( x)

Proof of Theorem mpteq12dva

Step	Hyp	Ref	Expression
1		mpteq12dv.1	. . 3 |- ( ph -> A = C )
2	1	alrimiv 1796	. 2 |- ( ph -> A. x A = C )
3		mpteq12dva.2	. . 3 |- ( ( ph /\ x e. A ) -> B = D )
4	3	ralrimiva 2435	. 2 |- ( ph -> A. x e. A B = D )
5		mpteq12f 3860	. 2 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )
6	2, 4, 5	syl2anc 403	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1283   = wceq 1285   e. wcel 1434   A.wral 2349   |-> cmpt 3841

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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-ral 2354  df-opab 3842  df-mpt 3843

This theorem is referenced by:  mpteq12dv  3862