Theorem bj-nnfa1

Description: See nfa1 . (Contributed by BJ, 12-Aug-2023) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-nnfa1	$$⊢Ⅎ\text{'}x\forall x\phi$$

Proof

Step	Hyp	Ref	Expression
1		hbe1a	$$⊢\exists x\forall x\phi \to \forall x\phi$$
2		bj-modal4	$$⊢\forall x\phi \to \forall x\forall x\phi$$
3		df-bj-nnf	$$⊢Ⅎ\text{'}x\forall x\phi \leftrightarrow \left(\left(\exists x\forall x\phi \to \forall x\phi \right)\wedge \left(\forall x\phi \to \forall x\forall x\phi \right)\right)$$
4	1 2 3	mpbir2an	$$⊢Ⅎ\text{'}x\forall x\phi$$