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Metamath Proof Explorer

Theorem mpteq12dva

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017) Remove dependency on ax-10 , ax-12 . (Revised by SN, 11-Nov-2024)

		Ref	Expression
	Hypotheses	mpteq12dv.1	$⊢ φ \to A = C$
		mpteq12dva.2	$⊢ (φ \land x \in A) \to B = D$
	Assertion	mpteq12dva	$⊢ φ \to (x \in A ⟼ B) = (x \in C ⟼ D)$

Proof

Step	Hyp	Ref	Expression
1		mpteq12dv.1	$⊢ φ \to A = C$
2		mpteq12dva.2	$⊢ (φ \land x \in A) \to B = D$
3	2	eqeq2d	$⊢ (φ \land x \in A) \to (y = B \leftrightarrow y = D)$
4	3	pm5.32da	$⊢ φ \to ((x \in A \land y = B) \leftrightarrow (x \in A \land y = D))$
5	1	eleq2d	$⊢ φ \to (x \in A \leftrightarrow x \in C)$
6	5	anbi1d	$⊢ φ \to ((x \in A \land y = D) \leftrightarrow (x \in C \land y = D))$
7	4 6	bitrd	$⊢ φ \to ((x \in A \land y = B) \leftrightarrow (x \in C \land y = D))$
8	7	opabbidv	$⊢ φ \to \{⟨x, y⟩ \| (x \in A \land y = B)\} = \{⟨x, y⟩ \| (x \in C \land y = D)\}$
9		df-mpt	$⊢ (x \in A ⟼ B) = \{⟨x, y⟩ \| (x \in A \land y = B)\}$
10		df-mpt	$⊢ (x \in C ⟼ D) = \{⟨x, y⟩ \| (x \in C \land y = D)\}$
11	8 9 10	3eqtr4g	$⊢ φ \to (x \in A ⟼ B) = (x \in C ⟼ D)$