Theorem cnf2dd 35361

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

Hypotheses

Ref	Expression
cnf2dd.1	⊢ (𝜑 → (𝜓 → ¬ 𝜃))
cnf2dd.2	⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))

Assertion

Ref	Expression
cnf2dd	⊢ (𝜑 → (𝜓 → 𝜒))

Proof of Theorem cnf2dd

Step	Hyp	Ref	Expression
1		cnf2dd.1	. 2 ⊢ (𝜑 → (𝜓 → ¬ 𝜃))
2		cnf2dd.2	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))
3		pm1.4 865	. . 3 ⊢ ((𝜒 ∨ 𝜃) → (𝜃 ∨ 𝜒))
4	2, 3	syl6 35	. 2 ⊢ (𝜑 → (𝜓 → (𝜃 ∨ 𝜒)))
5	1, 4	cnf1dd 35360	1 ⊢ (𝜑 → (𝜓 → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844

This theorem is referenced by:  cnfn2dd  35363  mpobi123f  35432  mptbi12f  35436  ac6s6  35442