Theorem bj-alnnf2

Description: If a proposition holds, then it holds for all values of a given variable if and only if it does not depend on that variable. (Contributed by BJ, 28-Mar-2026)

	Ref	Expression
Assertion	bj-alnnf2	$$⊢\phi \to \left(\forall x\phi \leftrightarrow Ⅎ\text{'}x\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-alnnf	$$⊢\left(\phi \to \forall x\phi \right)\leftrightarrow \left(\phi \to Ⅎ\text{'}x\phi \right)$$
2	1	pm5.74ri	$$⊢\phi \to \left(\forall x\phi \leftrightarrow Ⅎ\text{'}x\phi \right)$$