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Metamath Proof Explorer

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 GG, 1-Dec-2023)

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

Proof

Step	Hyp	Ref	Expression
1		mpteq12dv.1	$⊢ φ \to A = C$
2		mpteq12dv.2	$⊢ φ \to B = D$
3	2	adantr	$⊢ (φ \land x \in A) \to B = D$
4	1 3	mpteq12dva	$⊢ φ \to (x \in A ⟼ B) = (x \in C ⟼ D)$