Theorem bj-nnford

Description: Nonfreeness in both disjuncts implies nonfreeness in the disjunction, deduction form. See comments for bj-nnfor and bj-nnfand . (Contributed by BJ, 2-Dec-2023) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		bj-nnford.1	$$⊢\phi \to Ⅎ\text{'}x\psi$$
2		bj-nnford.2	$$⊢\phi \to Ⅎ\text{'}x\chi$$
3		19.43	$$⊢\exists x\left(\psi \vee \chi \right)\leftrightarrow \left(\exists x\psi \vee \exists x\chi \right)$$
4	1	bj-nnfed	$$⊢\phi \to \left(\exists x\psi \to \psi \right)$$
5	2	bj-nnfed	$$⊢\phi \to \left(\exists x\chi \to \chi \right)$$
6	4 5	orim12d	$$⊢\phi \to \left(\left(\exists x\psi \vee \exists x\chi \right)\to \left(\psi \vee \chi \right)\right)$$
7	3 6	biimtrid	$$⊢\phi \to \left(\exists x\left(\psi \vee \chi \right)\to \left(\psi \vee \chi \right)\right)$$
8	1	bj-nnfad	$$⊢\phi \to \left(\psi \to \forall x\psi \right)$$
9	2	bj-nnfad	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
10	8 9	orim12d	$$⊢\phi \to \left(\left(\psi \vee \chi \right)\to \left(\forall x\psi \vee \forall x\chi \right)\right)$$
11		19.33	$$⊢\left(\forall x\psi \vee \forall x\chi \right)\to \forall x\left(\psi \vee \chi \right)$$
12	10 11	syl6	$$⊢\phi \to \left(\left(\psi \vee \chi \right)\to \forall x\left(\psi \vee \chi \right)\right)$$
13		df-bj-nnf	$$⊢Ⅎ\text{'}x\left(\psi \vee \chi \right)\leftrightarrow \left(\left(\exists x\left(\psi \vee \chi \right)\to \left(\psi \vee \chi \right)\right)\wedge \left(\left(\psi \vee \chi \right)\to \forall x\left(\psi \vee \chi \right)\right)\right)$$
14	7 12 13	sylanbrc	$$⊢\phi \to Ⅎ\text{'}x\left(\psi \vee \chi \right)$$