Theorem iineq2dv

Description: Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004)

	Ref	Expression
Hypothesis	iuneq2dv.1	|- ( ( ph /\ x e. A ) -> B = C )
Assertion	iineq2dv	|- ( ph -> |^|_ x e. A B = |^|_ x e. A C )

Proof

Step	Hyp	Ref	Expression
1		iuneq2dv.1	|- ( ( ph /\ x e. A ) -> B = C )
2	1	ralrimiva	|- ( ph -> A. x e. A B = C )
3		iineq2	|- ( A. x e. A B = C -> |^|_ x e. A B = |^|_ x e. A C )
4	2 3	syl	|- ( ph -> |^|_ x e. A B = |^|_ x e. A C )