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Metamath Proof Explorer

Theorem ixpeq2dva

Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016)

		Ref	Expression
	Hypothesis	ixpeq2dva.1	$⊢ (φ \land x \in A) \to B = C$
	Assertion	ixpeq2dva	$⊢ φ \to \underset{x \in A}{⨉} B = \underset{x \in A}{⨉} C$

Step	Hyp	Ref	Expression
1		ixpeq2dva.1	$⊢ (φ \land x \in A) \to B = C$
2	1	ralrimiva	$⊢ φ \to \forall x \in A B = C$
3		ixpeq2	$⊢ \forall x \in A B = C \to \underset{x \in A}{⨉} B = \underset{x \in A}{⨉} C$
4	2 3	syl	$⊢ φ \to \underset{x \in A}{⨉} B = \underset{x \in A}{⨉} C$