Theorem ixpeq2d

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

	Ref	Expression
Hypotheses	ixpeq2d.1	|- F/ x ph
	ixpeq2d.2	|- ( ( ph /\ x e. A ) -> B = C )
Assertion	ixpeq2d	|- ( ph -> X_ x e. A B = X_ x e. A C )

Proof

Step	Hyp	Ref	Expression
1		ixpeq2d.1	|- F/ x ph
2		ixpeq2d.2	|- ( ( ph /\ x e. A ) -> B = C )
3	2	ex	|- ( ph -> ( x e. A -> B = C ) )
4	1 3	ralrimi	|- ( ph -> A. x e. A B = C )
5		ixpeq2	|- ( A. x e. A B = C -> X_ x e. A B = X_ x e. A C )
6	4 5	syl	|- ( ph -> X_ x e. A B = X_ x e. A C )