Theorem cnfn2dd 36178

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

Hypotheses

Ref	Expression
cnfn2dd.1	⊢ (𝜑 → (𝜓 → 𝜃))
cnfn2dd.2	⊢ (𝜑 → (𝜓 → (𝜒 ∨ ¬ 𝜃)))

Assertion

Ref	Expression
cnfn2dd	⊢ (𝜑 → (𝜓 → 𝜒))

Proof of Theorem cnfn2dd

Step	Hyp	Ref	Expression
1		cnfn2dd.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜃))
2		notnot 142	. . 3 ⊢ (𝜃 → ¬ ¬ 𝜃)
3	1, 2	syl6 35	. 2 ⊢ (𝜑 → (𝜓 → ¬ ¬ 𝜃))
4		cnfn2dd.2	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 ∨ ¬ 𝜃)))
5	3, 4	cnf2dd 36176	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 206  df-an 396  df-or 844

This theorem is referenced by:  mpobi123f  36247  mptbi12f  36251  ac6s6  36257