Theorem mpteq12dva

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

	Ref	Expression
Hypotheses	mpteq12dv.1	$$⊢\phi \to A=C$$
	mpteq12dva.2	$$⊢\left(\phi \wedge x\in A\right)\to B=D$$
Assertion	mpteq12dva	$$⊢\phi \to \left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$

Proof

Step	Hyp	Ref	Expression
1		mpteq12dv.1	$$⊢\phi \to A=C$$
2		mpteq12dva.2	$$⊢\left(\phi \wedge x\in A\right)\to B=D$$
3	1	alrimiv	$$⊢\phi \to \forall xA=C$$
4	2	ralrimiva	$$⊢\phi \to \forall x\in AB=D$$
5		mpteq12f	$$⊢\left(\forall xA=C\wedge \forall x\in AB=D\right)\to \left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$
6	3 4 5	syl2anc	$$⊢\phi \to \left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$