Theorem pm4.82 894

Description: Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm4.82	⊢ (((φ → ψ) ∧ (φ → ¬ ψ)) ↔ ¬ φ)

Proof of Theorem pm4.82

Step	Hyp	Ref	Expression
1		pm2.65 164	. . 3 ⊢ ((φ → ψ) → ((φ → ¬ ψ) → ¬ φ))
2	1	imp 418	. 2 ⊢ (((φ → ψ) ∧ (φ → ¬ ψ)) → ¬ φ)
3		pm2.21 100	. . 3 ⊢ (¬ φ → (φ → ψ))
4		pm2.21 100	. . 3 ⊢ (¬ φ → (φ → ¬ ψ))
5	3, 4	jca 518	. 2 ⊢ (¬ φ → ((φ → ψ) ∧ (φ → ¬ ψ)))
6	2, 5	impbii 180	1 ⊢ (((φ → ψ) ∧ (φ → ¬ ψ)) ↔ ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358

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-an 360

This theorem is referenced by: (None)