Theorem bnj1212

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

	Ref	Expression
Hypotheses	bnj1212.1	$$⊢B=\left\{x\in A|\phi \right\}$$
	bnj1212.2	$$⊢\theta \leftrightarrow \left(\chi \wedge x\in B\wedge \tau \right)$$
Assertion	bnj1212	$$⊢\theta \to x\in A$$

Proof

Step	Hyp	Ref	Expression
1		bnj1212.1	$$⊢B=\left\{x\in A|\phi \right\}$$
2		bnj1212.2	$$⊢\theta \leftrightarrow \left(\chi \wedge x\in B\wedge \tau \right)$$
3	1	ssrab3	$$⊢B\subseteq A$$
4	2	simp2bi	$$⊢\theta \to x\in B$$
5	3 4	bnj1213	$$⊢\theta \to x\in A$$