Theorem bj-nnfe1

Description: See nfe1 . (Contributed by BJ, 12-Aug-2023) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-nnfe1	$$⊢Ⅎ\text{'}x\exists x\phi$$

Proof

Step	Hyp	Ref	Expression
1		bj-modal4e	$$⊢\exists x\exists x\phi \to \exists x\phi$$
2		hbe1	$$⊢\exists x\phi \to \forall x\exists x\phi$$
3		df-bj-nnf	$$⊢Ⅎ\text{'}x\exists x\phi \leftrightarrow \left(\left(\exists x\exists x\phi \to \exists x\phi \right)\wedge \left(\exists x\phi \to \forall x\exists x\phi \right)\right)$$
4	1 2 3	mpbir2an	$$⊢Ⅎ\text{'}x\exists x\phi$$