Theorem iineq2d

Description: Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011)

	Ref	Expression
Hypotheses	iineq2d.1	|- F/ x ph
	iineq2d.2	|- ( ( ph /\ x e. A ) -> B = C )
Assertion	iineq2d	|- ( ph -> |^|_ x e. A B = |^|_ x e. A C )

Proof

Step	Hyp	Ref	Expression
1		iineq2d.1	|- F/ x ph
2		iineq2d.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		iineq2	|- ( A. x e. A B = C -> |^|_ x e. A B = |^|_ x e. A C )
6	4 5	syl	|- ( ph -> |^|_ x e. A B = |^|_ x e. A C )