Theorem mpteq12da

Description: An equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	mpteq12da.1	|- F/ x ph
	mpteq12da.2	|- ( ph -> A = C )
	mpteq12da.3	|- ( ( ph /\ x e. A ) -> B = D )
Assertion	mpteq12da	|- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Proof

Step	Hyp	Ref	Expression
1		mpteq12da.1	|- F/ x ph
2		mpteq12da.2	|- ( ph -> A = C )
3		mpteq12da.3	|- ( ( ph /\ x e. A ) -> B = D )
4	1 2	alrimi	|- ( ph -> A. x A = C )
5	1 3	ralrimia	|- ( ph -> A. x e. A B = D )
6		mpteq12f	|- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )
7	4 5 6	syl2anc	|- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )