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Metamath Proof Explorer

Theorem iinin1

Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in Enderton p. 30. Use intiin to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015)

		Ref	Expression
	Assertion	iinin1	$⊢ A \neq \emptyset \to ⋂_{x \in A} (C \cap B) = ⋂_{x \in A} C \cap B$

Proof

Step	Hyp	Ref	Expression
1		iinin2	$⊢ A \neq \emptyset \to ⋂_{x \in A} (B \cap C) = B \cap ⋂_{x \in A} C$
2		incom	$⊢ C \cap B = B \cap C$
3	2	a1i	$⊢ x \in A \to C \cap B = B \cap C$
4	3	iineq2i	$⊢ ⋂_{x \in A} (C \cap B) = ⋂_{x \in A} (B \cap C)$
5		incom	$⊢ ⋂_{x \in A} C \cap B = B \cap ⋂_{x \in A} C$
6	1 4 5	3eqtr4g	$⊢ A \neq \emptyset \to ⋂_{x \in A} (C \cap B) = ⋂_{x \in A} C \cap B$