Theorem mpteq12dva 4124

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 1896	. 2 |- ( ph -> A. x A = C )
3		mpteq12dva.2	. . 3 |- ( ( ph /\ x e. A ) -> B = D )
4	3	ralrimiva 2578	. 2 |- ( ph -> A. x e. A B = D )
5		mpteq12f 4123	. 2 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )
6	2, 4, 5	syl2anc 411	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1370   = wceq 1372   e. wcel 2175   A.wral 2483   |-> cmpt 4104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-ral 2488  df-opab 4105  df-mpt 4106

This theorem is referenced by:  mpteq12dv  4125