Theorem bj-nnfimd

Description: Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication, deduction form. (Contributed by BJ, 2-Dec-2023)

	Ref	Expression
Hypotheses	bj-nnfimd.1	$$⊢\phi \to Ⅎ\text{'}x\psi$$
	bj-nnfimd.2	$$⊢\phi \to Ⅎ\text{'}x\chi$$
Assertion	bj-nnfimd	$$⊢\phi \to Ⅎ\text{'}x\left(\psi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-nnfimd.1	$$⊢\phi \to Ⅎ\text{'}x\psi$$
2		bj-nnfimd.2	$$⊢\phi \to Ⅎ\text{'}x\chi$$
3		bj-nnfim	$$⊢\left(Ⅎ\text{'}x\psi \wedge Ⅎ\text{'}x\chi \right)\to Ⅎ\text{'}x\left(\psi \to \chi \right)$$
4	1 2 3	syl2anc	$$⊢\phi \to Ⅎ\text{'}x\left(\psi \to \chi \right)$$