Theorem elrab3

Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006)

	Ref	Expression
Hypothesis	elrab.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	elrab3	$$⊢A\in B\to \left(A\in \left\{x\in B|\phi \right\}\leftrightarrow \psi \right)$$

Proof

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