Theorem 3ori 1419

Description: Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.)

Hypothesis

Ref	Expression
3ori.1	⊢ (𝜑 ∨ 𝜓 ∨ 𝜒)

Assertion

Ref	Expression
3ori	⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)

Proof of Theorem 3ori

Step	Hyp	Ref	Expression
1		ioran 980	. 2 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
2		3ori.1	. . . 4 ⊢ (𝜑 ∨ 𝜓 ∨ 𝜒)
3		df-3or 1083	. . . 4 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
4	2, 3	mpbi 232	. . 3 ⊢ ((𝜑 ∨ 𝜓) ∨ 𝜒)
5	4	ori 857	. 2 ⊢ (¬ (𝜑 ∨ 𝜓) → 𝜒)
6	1, 5	sylbir 237	1 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   ∨ w3o 1081

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  df-3or 1083

This theorem is referenced by:  rankxplim3  9302