Theorem pm4.82 1041

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		pm2.65 195	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑))
2	1	imp 412	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) → ¬ 𝜑)
3		pm2.21 124	. . 3 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
4		pm2.21 124	. . 3 ⊢ (¬ 𝜑 → (𝜑 → ¬ 𝜓))
5	3, 4	jca 521	. 2 ⊢ (¬ 𝜑 → ((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)))
6	2, 5	impbii 212	1 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  alimp-no-surprise  50618