Theorem bnj596

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

	Ref	Expression
Hypotheses	bnj596.1	$$⊢\phi \to \forall x\phi$$
	bnj596.2	$$⊢\phi \to \exists x\psi$$
Assertion	bnj596	$$⊢\phi \to \exists x\left(\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bnj596.1	$$⊢\phi \to \forall x\phi$$
2		bnj596.2	$$⊢\phi \to \exists x\psi$$
3	2	ancli	$$⊢\phi \to \left(\phi \wedge \exists x\psi \right)$$
4	1	nf5i	$$⊢Ⅎx\phi$$
5	4	19.42	$$⊢\exists x\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \exists x\psi \right)$$
6	3 5	sylibr	$$⊢\phi \to \exists x\left(\phi \wedge \psi \right)$$