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Metamath Proof Explorer

Theorem iineq2dv

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

		Ref	Expression
	Hypothesis	iuneq2dv.1	$⊢ (φ \land x \in A) \to B = C$
	Assertion	iineq2dv	$⊢ φ \to ⋂_{x \in A} B = ⋂_{x \in A} C$

Step	Hyp	Ref	Expression
1		iuneq2dv.1	$⊢ (φ \land x \in A) \to B = C$
2	1	ralrimiva	$⊢ φ \to \forall x \in A B = C$
3		iineq2	$⊢ \forall x \in A B = C \to ⋂_{x \in A} B = ⋂_{x \in A} C$
4	2 3	syl	$⊢ φ \to ⋂_{x \in A} B = ⋂_{x \in A} C$