Theorem ixpeq2d

Description: Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020)

	Ref	Expression
Hypotheses	ixpeq2d.1	$$⊢Ⅎx\phi$$
	ixpeq2d.2	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
Assertion	ixpeq2d	$$⊢\phi \to \underset{x\in A}{⨉}B=\underset{x\in A}{⨉}C$$

Proof

Step	Hyp	Ref	Expression
1		ixpeq2d.1	$$⊢Ⅎx\phi$$
2		ixpeq2d.2	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
3	2	ex	$$⊢\phi \to \left(x\in A\to B=C\right)$$
4	1 3	ralrimi	$$⊢\phi \to \forall x\in AB=C$$
5		ixpeq2	$$⊢\forall x\in AB=C\to \underset{x\in A}{⨉}B=\underset{x\in A}{⨉}C$$
6	4 5	syl	$$⊢\phi \to \underset{x\in A}{⨉}B=\underset{x\in A}{⨉}C$$