Theorem bnj1340

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

	Ref	Expression
Hypotheses	bnj1340.1	$$⊢\psi \to \exists x\theta$$
	bnj1340.2	$$⊢\chi \leftrightarrow \left(\psi \wedge \theta \right)$$
	bnj1340.3	$$⊢\psi \to \forall x\psi$$
Assertion	bnj1340	$$⊢\psi \to \exists x\chi$$

Proof

Step	Hyp	Ref	Expression
1		bnj1340.1	$$⊢\psi \to \exists x\theta$$
2		bnj1340.2	$$⊢\chi \leftrightarrow \left(\psi \wedge \theta \right)$$
3		bnj1340.3	$$⊢\psi \to \forall x\psi$$
4	3 1	bnj596	$$⊢\psi \to \exists x\left(\psi \wedge \theta \right)$$
5	4 2	bnj1198	$$⊢\psi \to \exists x\chi$$