Theorem cnf1dd

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

	Ref	Expression
Hypotheses	cnf1dd.1	$$⊢\phi \to \left(\psi \to \neg \chi \right)$$
	cnf1dd.2	$$⊢\phi \to \left(\psi \to \left(\chi \vee \theta \right)\right)$$
Assertion	cnf1dd	$$⊢\phi \to \left(\psi \to \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		cnf1dd.1	$$⊢\phi \to \left(\psi \to \neg \chi \right)$$
2		cnf1dd.2	$$⊢\phi \to \left(\psi \to \left(\chi \vee \theta \right)\right)$$
3	1 2	jcad	$$⊢\phi \to \left(\psi \to \left(\neg \chi \wedge \left(\chi \vee \theta \right)\right)\right)$$
4		df-or	$$⊢\left(\chi \vee \theta \right)\leftrightarrow \left(\neg \chi \to \theta \right)$$
5		pm3.35	$$⊢\left(\neg \chi \wedge \left(\neg \chi \to \theta \right)\right)\to \theta$$
6	4 5	sylan2b	$$⊢\left(\neg \chi \wedge \left(\chi \vee \theta \right)\right)\to \theta$$
7	3 6	syl6	$$⊢\phi \to \left(\psi \to \theta \right)$$