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Metamath Proof Explorer

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	$⊢ Ⅎ x φ$
	mpteq12df.2	$⊢ φ \to A = C$
	mpteq12df.3	$⊢ φ \to B = D$
Assertion	mpteq12df	$⊢ φ \to (x \in A ⟼ B) = (x \in C ⟼ D)$

Proof

Step	Hyp	Ref	Expression
1		mpteq12df.1	$⊢ Ⅎ x φ$
2		mpteq12df.2	$⊢ φ \to A = C$
3		mpteq12df.3	$⊢ φ \to B = D$
4	3	adantr	$⊢ (φ \land x \in A) \to B = D$
5	1 2 4	mpteq12da	$⊢ φ \to (x \in A ⟼ B) = (x \in C ⟼ D)$