Theorem pm2.82 802

Description: Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm2.82	⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) → (((𝜑 ∨ ¬ 𝜒) ∨ 𝜃) → ((𝜑 ∨ 𝜓) ∨ 𝜃)))

Proof of Theorem pm2.82

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ ((𝜑 ∨ 𝜓) → ((𝜑 ∨ ¬ 𝜒) → (𝜑 ∨ 𝜓)))
2		pm2.24 611	. . . 4 ⊢ (𝜒 → (¬ 𝜒 → 𝜓))
3	2	orim2d 778	. . 3 ⊢ (𝜒 → ((𝜑 ∨ ¬ 𝜒) → (𝜑 ∨ 𝜓)))
4	1, 3	jaoi 706	. 2 ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) → ((𝜑 ∨ ¬ 𝜒) → (𝜑 ∨ 𝜓)))
5	4	orim1d 777	1 ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) → (((𝜑 ∨ ¬ 𝜒) ∨ 𝜃) → ((𝜑 ∨ 𝜓) ∨ 𝜃)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)