Theorem pm4.82 891

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		pm2.65 617	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑))
2	1	imp 122	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) → ¬ 𝜑)
3		pm2.21 579	. . 3 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
4		pm2.21 579	. . 3 ⊢ (¬ 𝜑 → (𝜑 → ¬ 𝜓))
5	3, 4	jca 300	. 2 ⊢ (¬ 𝜑 → ((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)))
6	2, 5	impbii 124	1 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)