Theorem mpteq2da

Description: Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013) (Revised by Mario Carneiro, 16-Dec-2013) (Proof shortened by SN, 11-Nov-2024)

	Ref	Expression
Hypotheses	mpteq2da.1	$$⊢Ⅎx\phi$$
	mpteq2da.2	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
Assertion	mpteq2da	$$⊢\phi \to \left(x\in A⟼B\right)=\left(x\in A⟼C\right)$$

Proof

Step	Hyp	Ref	Expression
1		mpteq2da.1	$$⊢Ⅎx\phi$$
2		mpteq2da.2	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
3		eqidd	$$⊢\phi \to A=A$$
4	1 3 2	mpteq12da	$$⊢\phi \to \left(x\in A⟼B\right)=\left(x\in A⟼C\right)$$