Theorem mpteq12da

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

	Ref	Expression
Hypotheses	mpteq12da.1	$$⊢Ⅎx\phi$$
	mpteq12da.2	$$⊢\phi \to A=C$$
	mpteq12da.3	$$⊢\left(\phi \wedge x\in A\right)\to B=D$$
Assertion	mpteq12da	$$⊢\phi \to \left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$

Proof

Step	Hyp	Ref	Expression
1		mpteq12da.1	$$⊢Ⅎx\phi$$
2		mpteq12da.2	$$⊢\phi \to A=C$$
3		mpteq12da.3	$$⊢\left(\phi \wedge x\in A\right)\to B=D$$
4	1 2	alrimi	$$⊢\phi \to \forall xA=C$$
5	1 3	ralrimia	$$⊢\phi \to \forall x\in AB=D$$
6		mpteq12f	$$⊢\left(\forall xA=C\wedge \forall x\in AB=D\right)\to \left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$
7	4 5 6	syl2anc	$$⊢\phi \to \left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$