Theorem bj-dfnnf3

Description: Alternate definition of nonfreeness when sp is available. (Contributed by BJ, 28-Jul-2023) The proof should not rely on df-nf . (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-dfnnf3	$$⊢Ⅎ\text{'}x\phi \leftrightarrow \left(\exists x\phi \to \forall x\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-nnfea	$$⊢Ⅎ\text{'}x\phi \to \left(\exists x\phi \to \forall x\phi \right)$$
2		bj-19.21bit	$$⊢\left(\exists x\phi \to \forall x\phi \right)\to \left(\exists x\phi \to \phi \right)$$
3		bj-19.23bit	$$⊢\left(\exists x\phi \to \forall x\phi \right)\to \left(\phi \to \forall x\phi \right)$$
4		df-bj-nnf	$$⊢Ⅎ\text{'}x\phi \leftrightarrow \left(\left(\exists x\phi \to \phi \right)\wedge \left(\phi \to \forall x\phi \right)\right)$$
5	2 3 4	sylanbrc	$$⊢\left(\exists x\phi \to \forall x\phi \right)\to Ⅎ\text{'}x\phi$$
6	1 5	impbii	$$⊢Ⅎ\text{'}x\phi \leftrightarrow \left(\exists x\phi \to \forall x\phi \right)$$