Theorem 3ori 1242

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 476	. 2 ⊢ (¬ (φ ∨ ψ) ↔ (¬ φ ∧ ¬ ψ))
2		3ori.1	. . . 4 ⊢ (φ ∨ ψ ∨ χ)
3		df-3or 935	. . . 4 ⊢ ((φ ∨ ψ ∨ χ) ↔ ((φ ∨ ψ) ∨ χ))
4	2, 3	mpbi 199	. . 3 ⊢ ((φ ∨ ψ) ∨ χ)
5	4	ori 364	. 2 ⊢ (¬ (φ ∨ ψ) → χ)
6	1, 5	sylbir 204	1 ⊢ ((¬ φ ∧ ¬ ψ) → χ)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   ∨ w3o 933

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 177  df-or 359  df-an 360  df-3or 935

This theorem is referenced by: (None)