Theorem cnfn2dd

Description: A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018)

	Ref	Expression
Hypotheses	cnfn2dd.1	$$⊢\phi \to \left(\psi \to \theta \right)$$
	cnfn2dd.2	$$⊢\phi \to \left(\psi \to \left(\chi \vee \neg \theta \right)\right)$$
Assertion	cnfn2dd	$$⊢\phi \to \left(\psi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cnfn2dd.1	$$⊢\phi \to \left(\psi \to \theta \right)$$
2		cnfn2dd.2	$$⊢\phi \to \left(\psi \to \left(\chi \vee \neg \theta \right)\right)$$
3		notnot	$$⊢\theta \to \neg \neg \theta$$
4	1 3	syl6	$$⊢\phi \to \left(\psi \to \neg \neg \theta \right)$$
5	4 2	cnf2dd	$$⊢\phi \to \left(\psi \to \chi \right)$$