Theorem bnj1345

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

	Ref	Expression
Hypotheses	bnj1345.1	$$⊢\phi \to \exists x\left(\psi \wedge \chi \right)$$
	bnj1345.2	$$⊢\theta \leftrightarrow \left(\phi \wedge \psi \wedge \chi \right)$$
	bnj1345.3	$$⊢\phi \to \forall x\phi$$
Assertion	bnj1345	$$⊢\phi \to \exists x\theta$$

Proof

Step	Hyp	Ref	Expression
1		bnj1345.1	$$⊢\phi \to \exists x\left(\psi \wedge \chi \right)$$
2		bnj1345.2	$$⊢\theta \leftrightarrow \left(\phi \wedge \psi \wedge \chi \right)$$
3		bnj1345.3	$$⊢\phi \to \forall x\phi$$
4	1 3	bnj1275	$$⊢\phi \to \exists x\left(\phi \wedge \psi \wedge \chi \right)$$
5	4 2	bnj1198	$$⊢\phi \to \exists x\theta$$