Theorem bj-nfnnfTEMP

Description: New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp . (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-nfnnfTEMP	$$⊢Ⅎ\text{'}x\phi \leftrightarrow Ⅎx\phi$$

Proof

Step	Hyp	Ref	Expression
1		bj-dfnnf3	$$⊢Ⅎ\text{'}x\phi \leftrightarrow \left(\exists x\phi \to \forall x\phi \right)$$
2		df-nf	$$⊢Ⅎx\phi \leftrightarrow \left(\exists x\phi \to \forall x\phi \right)$$
3	1 2	bitr4i	$$⊢Ⅎ\text{'}x\phi \leftrightarrow Ⅎx\phi$$