Theorem ineqri

Description: Inference from membership to intersection. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Hypothesis	ineqri.1	$$⊢\left(x\in A\wedge x\in B\right)\leftrightarrow x\in C$$
Assertion	ineqri	$$⊢A\cap B=C$$

Proof

Step	Hyp	Ref	Expression
1		ineqri.1	$$⊢\left(x\in A\wedge x\in B\right)\leftrightarrow x\in C$$
2		elin	$$⊢x\in \left(A\cap B\right)\leftrightarrow \left(x\in A\wedge x\in B\right)$$
3	2 1	bitri	$$⊢x\in \left(A\cap B\right)\leftrightarrow x\in C$$
4	3	eqriv	$$⊢A\cap B=C$$