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	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	cnfn1dd.2	⊢ ( 𝜑 → ( 𝜓 → ( ¬ 𝜒 ∨ 𝜃 ) ) )
Assertion	cnfn1dd	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		cnfn1dd.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		cnfn1dd.2	⊢ ( 𝜑 → ( 𝜓 → ( ¬ 𝜒 ∨ 𝜃 ) ) )
3		notnot	⊢ ( 𝜒 → ¬ ¬ 𝜒 )
4	1 3	syl6	⊢ ( 𝜑 → ( 𝜓 → ¬ ¬ 𝜒 ) )
5	4 2	cnf1dd	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )