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Metamath Proof Explorer

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 )