Theorem pm4.82 945

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		pm2.65 654	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑))
2	1	imp 123	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) → ¬ 𝜑)
3		pm2.21 612	. . 3 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
4		pm2.21 612	. . 3 ⊢ (¬ 𝜑 → (𝜑 → ¬ 𝜓))
5	3, 4	jca 304	. 2 ⊢ (¬ 𝜑 → ((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)))
6	2, 5	impbii 125	1 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104

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-in1 609  ax-in2 610

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)