Theorem bj-nnfnt

Description: A variable is nonfree in a formula if and only if it is nonfree in its negation. The foward implication is intuitionistically valid (and that direction is sufficient for the purpose of recursively proving that some formulas have a given variable not free in them, like bj-nnfim ). Intuitionistically, |- ( F// x -. ph <-> F// x -. -. ph ) . See nfnt . (Contributed by BJ, 28-Jul-2023)

	Ref	Expression
Assertion	bj-nnfnt	$$⊢Ⅎ\text{'}x\phi \leftrightarrow Ⅎ\text{'}x\neg \phi$$

Proof

Step	Hyp	Ref	Expression
1		eximal	$$⊢\left(\exists x\phi \to \phi \right)\leftrightarrow \left(\neg \phi \to \forall x\neg \phi \right)$$
2		alimex	$$⊢\left(\phi \to \forall x\phi \right)\leftrightarrow \left(\exists x\neg \phi \to \neg \phi \right)$$
3	1 2	anbi12ci	$$⊢\left(\left(\exists x\phi \to \phi \right)\wedge \left(\phi \to \forall x\phi \right)\right)\leftrightarrow \left(\left(\exists x\neg \phi \to \neg \phi \right)\wedge \left(\neg \phi \to \forall x\neg \phi \right)\right)$$
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		df-bj-nnf	$$⊢Ⅎ\text{'}x\neg \phi \leftrightarrow \left(\left(\exists x\neg \phi \to \neg \phi \right)\wedge \left(\neg \phi \to \forall x\neg \phi \right)\right)$$
6	3 4 5	3bitr4i	$$⊢Ⅎ\text{'}x\phi \leftrightarrow Ⅎ\text{'}x\neg \phi$$