Theorem elrab2

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006)

	Ref	Expression
Hypotheses	elrab2.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	elrab2.2	$$⊢C=\left\{x\in B|\phi \right\}$$
Assertion	elrab2	$$⊢A\in C\leftrightarrow \left(A\in B\wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		elrab2.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		elrab2.2	$$⊢C=\left\{x\in B|\phi \right\}$$
3	2	eleq2i	$$⊢A\in C\leftrightarrow A\in \left\{x\in B|\phi \right\}$$
4	1	elrab	$$⊢A\in \left\{x\in B|\phi \right\}\leftrightarrow \left(A\in B\wedge \psi \right)$$
5	3 4	bitri	$$⊢A\in C\leftrightarrow \left(A\in B\wedge \psi \right)$$