Theorem rabbi2dva

Description: Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014)

	Ref	Expression
Hypothesis	rabbi2dva.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(x\in B\leftrightarrow \psi \right)$$
Assertion	rabbi2dva	$$⊢\phi \to A\cap B=\left\{x\in A|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		rabbi2dva.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(x\in B\leftrightarrow \psi \right)$$
2		dfin5	$$⊢A\cap B=\left\{x\in A|x\in B\right\}$$
3	1	rabbidva	$$⊢\phi \to \left\{x\in A|x\in B\right\}=\left\{x\in A|\psi \right\}$$
4	2 3	syl5eq	$$⊢\phi \to A\cap B=\left\{x\in A|\psi \right\}$$