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Metamath Proof Explorer

Theorem mpteq1

Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013) (Proof shortened by SN, 11-Nov-2024)

		Ref	Expression
	Assertion	mpteq1	$⊢ A = B \to (x \in A ⟼ C) = (x \in B ⟼ C)$

Step	Hyp	Ref	Expression
1		id	$⊢ A = B \to A = B$
2		eqidd	$⊢ A = B \to C = C$
3	1 2	mpteq12dv	$⊢ A = B \to (x \in A ⟼ C) = (x \in B ⟼ C)$