Theorem mpteq12i

Description: An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010) (Revised by Mario Carneiro, 16-Dec-2013)

	Ref	Expression
Hypotheses	mpteq12i.1	$$⊢A=C$$
	mpteq12i.2	$$⊢B=D$$
Assertion	mpteq12i	$$⊢\left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$

Proof

Step	Hyp	Ref	Expression
1		mpteq12i.1	$$⊢A=C$$
2		mpteq12i.2	$$⊢B=D$$
3	1	a1i	$$⊢\top \to A=C$$
4	2	a1i	$$⊢\top \to B=D$$
5	3 4	mpteq12dv	$$⊢\top \to \left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$
6	5	mptru	$$⊢\left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$