Theorem cnfn1dd

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

	Ref	Expression
Hypotheses	cnfn1dd.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	cnfn1dd.2	$$⊢\phi \to \left(\psi \to \left(\neg \chi \vee \theta \right)\right)$$
Assertion	cnfn1dd	$$⊢\phi \to \left(\psi \to \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		cnfn1dd.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		cnfn1dd.2	$$⊢\phi \to \left(\psi \to \left(\neg \chi \vee \theta \right)\right)$$
3		notnot	$$⊢\chi \to \neg \neg \chi$$
4	1 3	syl6	$$⊢\phi \to \left(\psi \to \neg \neg \chi \right)$$
5	4 2	cnf1dd	$$⊢\phi \to \left(\psi \to \theta \right)$$