Theorem pm4.82

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

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

Proof

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