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Metamath Proof Explorer

Theorem bj-nnfe1

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

		Ref	Expression
	Assertion	bj-nnfe1	$⊢ Ⅎ' x \exists x φ$

Step	Hyp	Ref	Expression
1		bj-modal4e	$⊢ \exists x \exists x φ \to \exists x φ$
2		hbe1	$⊢ \exists x φ \to \forall x \exists x φ$
3		df-bj-nnf	$⊢ Ⅎ' x \exists x φ \leftrightarrow ((\exists x \exists x φ \to \exists x φ) \land (\exists x φ \to \forall x \exists x φ))$
4	1 2 3	mpbir2an	$⊢ Ⅎ' x \exists x φ$