Theorem mpteq12dv

Description: An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011) (Revised by Mario Carneiro, 16-Dec-2013) Drop ax-10 while shortening its proof. (Revised by Steven Nguyen and Gino Giotto, 1-Dec-2023)

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

Proof

Step	Hyp	Ref	Expression
1		mpteq12dv.1	$$⊢\phi \to A=C$$
2		mpteq12dv.2	$$⊢\phi \to B=D$$
3		nfv	$$⊢Ⅎx\phi$$
4	3 1 2	mpteq12df	$$⊢\phi \to \left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$