Theorem ixpssixp

Description: Subclass theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 8-Apr-2021)

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

Proof

Step	Hyp	Ref	Expression
1		ixpssixp.1	$$⊢Ⅎx\phi$$
2		ixpssixp.2	$$⊢\left(\phi \wedge x\in A\right)\to B\subseteq C$$
3	2	ex	$$⊢\phi \to \left(x\in A\to B\subseteq C\right)$$
4	1 3	ralrimi	$$⊢\phi \to \forall x\in AB\subseteq C$$
5		ss2ixp	$$⊢\forall x\in AB\subseteq C\to \underset{x\in A}{⨉}B\subseteq \underset{x\in A}{⨉}C$$
6	4 5	syl	$$⊢\phi \to \underset{x\in A}{⨉}B\subseteq \underset{x\in A}{⨉}C$$