Theorem unitresl 867

Description: A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)

Hypotheses

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

Assertion

Ref	Expression
unitresl	⊢ (𝜑 → 𝜓)

Proof of Theorem unitresl

Step	Hyp	Ref	Expression
1		unitresl.1	. 2 ⊢ (𝜑 → (𝜓 ∨ 𝜒))
2		unitresl.2	. 2 ⊢ (𝜑 → ¬ 𝜒)
3		orcom 865	. . 3 ⊢ ((𝜓 ∨ 𝜒) ↔ (𝜒 ∨ 𝜓))
4		df-or 843	. . 3 ⊢ ((𝜒 ∨ 𝜓) ↔ (¬ 𝜒 → 𝜓))
5	3, 4	sylbb 220	. 2 ⊢ ((𝜓 ∨ 𝜒) → (¬ 𝜒 → 𝜓))
6	1, 2, 5	sylc 65	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 842

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 208  df-or 843

This theorem is referenced by:  unitresr  868