Theorem mpteq12df

Description: An equality inference for the maps-to notation. Compare mpteq12dv . (Contributed by Scott Fenton, 8-Aug-2013) (Revised by Mario Carneiro, 11-Dec-2016) (Proof shortened by SN, 11-Nov-2024)

	Ref	Expression
Hypotheses	mpteq12df.1	$$⊢Ⅎx\phi$$
	mpteq12df.2	$$⊢\phi \to A=C$$
	mpteq12df.3	$$⊢\phi \to B=D$$
Assertion	mpteq12df	$$⊢\phi \to \left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$

Proof

Step	Hyp	Ref	Expression
1		mpteq12df.1	$$⊢Ⅎx\phi$$
2		mpteq12df.2	$$⊢\phi \to A=C$$
3		mpteq12df.3	$$⊢\phi \to B=D$$
4	3	adantr	$$⊢\left(\phi \wedge x\in A\right)\to B=D$$
5	1 2 4	mpteq12da	$$⊢\phi \to \left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$