Theorem mpteq12dv

Description: An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011) (Revised by Mario Carneiro, 16-Dec-2013) Remove dependency on ax-10 , ax-12 . (Revised by SN 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	2	adantr	$$⊢\left(\phi \wedge x\in A\right)\to B=D$$
4	1 3	mpteq12dva	$$⊢\phi \to \left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$