Theorem ixpeq1d

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

	Ref	Expression
Hypothesis	ixpeq1d.1	$$⊢\phi \to A=B$$
Assertion	ixpeq1d	$$⊢\phi \to \underset{x\in A}{⨉}C=\underset{x\in B}{⨉}C$$

Proof

Step	Hyp	Ref	Expression
1		ixpeq1d.1	$$⊢\phi \to A=B$$
2		ixpeq1	$$⊢A=B\to \underset{x\in A}{⨉}C=\underset{x\in B}{⨉}C$$
3	1 2	syl	$$⊢\phi \to \underset{x\in A}{⨉}C=\underset{x\in B}{⨉}C$$