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	⊢ ( 𝜑 → ( 𝜓 → ¬ 𝜒 ) )
	cnf1dd.2	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ∨ 𝜃 ) ) )
Assertion	cnf1dd	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		cnf1dd.1	⊢ ( 𝜑 → ( 𝜓 → ¬ 𝜒 ) )
2		cnf1dd.2	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ∨ 𝜃 ) ) )
3	1 2	jcad	⊢ ( 𝜑 → ( 𝜓 → ( ¬ 𝜒 ∧ ( 𝜒 ∨ 𝜃 ) ) ) )
4		df-or	⊢ ( ( 𝜒 ∨ 𝜃 ) ↔ ( ¬ 𝜒 → 𝜃 ) )
5		pm3.35	⊢ ( ( ¬ 𝜒 ∧ ( ¬ 𝜒 → 𝜃 ) ) → 𝜃 )
6	4 5	sylan2b	⊢ ( ( ¬ 𝜒 ∧ ( 𝜒 ∨ 𝜃 ) ) → 𝜃 )
7	3 6	syl6	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )