Theorem bj-nnf-spim

Description: A universal specialization result in deduction form, proved from ax-1 -- ax-6 , where the only DV condition is on x , y and where x should be nonfree in the new proposition ch (and in the context ph ). (Contributed by BJ, 4-Apr-2026)

	Ref	Expression
Hypotheses	bj-nnf-spim.nf0	$$⊢\phi \to \forall x\phi$$
	bj-nnf-spim.nf	$$⊢\phi \to Ⅎ\text{'}x\chi$$
	bj-nnf-spim.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \to \chi \right)$$
Assertion	bj-nnf-spim	$$⊢\phi \to \left(\forall x\psi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-nnf-spim.nf0	$$⊢\phi \to \forall x\phi$$
2		bj-nnf-spim.nf	$$⊢\phi \to Ⅎ\text{'}x\chi$$
3		bj-nnf-spim.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \to \chi \right)$$
4	2	bj-nnfed	$$⊢\phi \to \left(\exists x\chi \to \chi \right)$$
5	1 4 3	bj-spim0	$$⊢\phi \to \left(\forall x\psi \to \chi \right)$$