Theorem rabeq2i

Description: Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004)

	Ref	Expression
Hypothesis	rabeq2i.1	$$⊢A=\left\{x\in B|\phi \right\}$$
Assertion	rabeq2i	$$⊢x\in A\leftrightarrow \left(x\in B\wedge \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		rabeq2i.1	$$⊢A=\left\{x\in B|\phi \right\}$$
2	1	eleq2i	$$⊢x\in A\leftrightarrow x\in \left\{x\in B|\phi \right\}$$
3		rabid	$$⊢x\in \left\{x\in B|\phi \right\}\leftrightarrow \left(x\in B\wedge \phi \right)$$
4	2 3	bitri	$$⊢x\in A\leftrightarrow \left(x\in B\wedge \phi \right)$$