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	⊢ ( 𝜑 → ( 𝜓 → ¬ 𝜃 ) )
	cnf2dd.2	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ∨ 𝜃 ) ) )
Assertion	cnf2dd	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cnf2dd.1	⊢ ( 𝜑 → ( 𝜓 → ¬ 𝜃 ) )
2		cnf2dd.2	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ∨ 𝜃 ) ) )
3		pm1.4	⊢ ( ( 𝜒 ∨ 𝜃 ) → ( 𝜃 ∨ 𝜒 ) )
4	2 3	syl6	⊢ ( 𝜑 → ( 𝜓 → ( 𝜃 ∨ 𝜒 ) ) )
5	1 4	cnf1dd	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )