Theorem ixpeq2dva

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

	Ref	Expression
Hypothesis	ixpeq2dva.1	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
Assertion	ixpeq2dva	$$⊢\phi \to \underset{x\in A}{⨉}B=\underset{x\in A}{⨉}C$$

Proof

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