Theorem bnj228

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (Proof shortened by Andrew Salmon, 9-Jul-2011) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bnj228.1	$$⊢\phi \leftrightarrow \forall x\in A\psi$$
Assertion	bnj228	$$⊢\left(x\in A\wedge \phi \right)\to \psi$$

Proof

Step	Hyp	Ref	Expression
1		bnj228.1	$$⊢\phi \leftrightarrow \forall x\in A\psi$$
2		rsp	$$⊢\forall x\in A\psi \to \left(x\in A\to \psi \right)$$
3	1 2	sylbi	$$⊢\phi \to \left(x\in A\to \psi \right)$$
4	3	impcom	$$⊢\left(x\in A\wedge \phi \right)\to \psi$$