Theorem cnf2dd

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

	Ref	Expression
Hypotheses	cnf2dd.1	$$⊢\phi \to \left(\psi \to \neg \theta \right)$$
	cnf2dd.2	$$⊢\phi \to \left(\psi \to \left(\chi \vee \theta \right)\right)$$
Assertion	cnf2dd	$$⊢\phi \to \left(\psi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cnf2dd.1	$$⊢\phi \to \left(\psi \to \neg \theta \right)$$
2		cnf2dd.2	$$⊢\phi \to \left(\psi \to \left(\chi \vee \theta \right)\right)$$
3		pm1.4	$$⊢\left(\chi \vee \theta \right)\to \left(\theta \vee \chi \right)$$
4	2 3	syl6	$$⊢\phi \to \left(\psi \to \left(\theta \vee \chi \right)\right)$$
5	1 4	cnf1dd	$$⊢\phi \to \left(\psi \to \chi \right)$$