Theorem bj-nnfnfTEMP

Description: New nonfreeness implies old nonfreeness on implicational calculus (the proof indicates it uses ax-3 because of set.mm's definition of the biconditional, but the proof actually holds in minimal implicational calculus). (Contributed by BJ, 28-Jul-2023) The proof should not rely on df-nf except via df-nf directly. (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-nnfnfTEMP	$$⊢Ⅎ\text{'}x\phi \to Ⅎx\phi$$

Proof

Step	Hyp	Ref	Expression
1		bj-nnfea	$$⊢Ⅎ\text{'}x\phi \to \left(\exists x\phi \to \forall x\phi \right)$$
2		df-nf	$$⊢Ⅎx\phi \leftrightarrow \left(\exists x\phi \to \forall x\phi \right)$$
3	1 2	sylibr	$$⊢Ⅎ\text{'}x\phi \to Ⅎx\phi$$