Theorem bnj1521

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

	Ref	Expression
Hypotheses	bnj1521.1	$$⊢\chi \to \exists x\in B\phi$$
	bnj1521.2	$$⊢\theta \leftrightarrow \left(\chi \wedge x\in B\wedge \phi \right)$$
	bnj1521.3	$$⊢\chi \to \forall x\chi$$
Assertion	bnj1521	$$⊢\chi \to \exists x\theta$$

Proof

Step	Hyp	Ref	Expression
1		bnj1521.1	$$⊢\chi \to \exists x\in B\phi$$
2		bnj1521.2	$$⊢\theta \leftrightarrow \left(\chi \wedge x\in B\wedge \phi \right)$$
3		bnj1521.3	$$⊢\chi \to \forall x\chi$$
4	1	bnj1196	$$⊢\chi \to \exists x\left(x\in B\wedge \phi \right)$$
5	4 2 3	bnj1345	$$⊢\chi \to \exists x\theta$$