Theorem ixpssixp

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

	Ref	Expression
Hypotheses	ixpssixp.1	|- F/ x ph
	ixpssixp.2	|- ( ( ph /\ x e. A ) -> B C_ C )
Assertion	ixpssixp	|- ( ph -> X_ x e. A B C_ X_ x e. A C )

Proof

Step	Hyp	Ref	Expression
1		ixpssixp.1	|- F/ x ph
2		ixpssixp.2	|- ( ( ph /\ x e. A ) -> B C_ C )
3	2	ex	|- ( ph -> ( x e. A -> B C_ C ) )
4	1 3	ralrimi	|- ( ph -> A. x e. A B C_ C )
5		ss2ixp	|- ( A. x e. A B C_ C -> X_ x e. A B C_ X_ x e. A C )
6	4 5	syl	|- ( ph -> X_ x e. A B C_ X_ x e. A C )