Theorem bj-alnnf

Description: In deduction-style proofs, it is equivalent to assert that the context holds for all values of a variable, or that is does not depend on that variable. (Contributed by BJ, 28-Mar-2026)

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

Proof

Step	Hyp	Ref	Expression
1		ax-1	$$⊢\phi \to \left(\exists x\phi \to \phi \right)$$
2	1	biantrur	$$⊢\left(\phi \to \left(\phi \to \forall x\phi \right)\right)\leftrightarrow \left(\left(\phi \to \left(\exists x\phi \to \phi \right)\right)\wedge \left(\phi \to \left(\phi \to \forall x\phi \right)\right)\right)$$
3		pm5.4	$$⊢\left(\phi \to \left(\phi \to \forall x\phi \right)\right)\leftrightarrow \left(\phi \to \forall x\phi \right)$$
4		pm4.76	$$⊢\left(\left(\phi \to \left(\exists x\phi \to \phi \right)\right)\wedge \left(\phi \to \left(\phi \to \forall x\phi \right)\right)\right)\leftrightarrow \left(\phi \to \left(\left(\exists x\phi \to \phi \right)\wedge \left(\phi \to \forall x\phi \right)\right)\right)$$
5	2 3 4	3bitr3i	$$⊢\left(\phi \to \forall x\phi \right)\leftrightarrow \left(\phi \to \left(\left(\exists x\phi \to \phi \right)\wedge \left(\phi \to \forall x\phi \right)\right)\right)$$
6		df-bj-nnf	$$⊢Ⅎ\text{'}x\phi \leftrightarrow \left(\left(\exists x\phi \to \phi \right)\wedge \left(\phi \to \forall x\phi \right)\right)$$
7	6	imbi2i	$$⊢\left(\phi \to Ⅎ\text{'}x\phi \right)\leftrightarrow \left(\phi \to \left(\left(\exists x\phi \to \phi \right)\wedge \left(\phi \to \forall x\phi \right)\right)\right)$$
8	5 7	bitr4i	$$⊢\left(\phi \to \forall x\phi \right)\leftrightarrow \left(\phi \to Ⅎ\text{'}x\phi \right)$$