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Metamath Proof Explorer

Theorem pm5.17

Description: Theorem *5.17 of WhiteheadRussell p. 124. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 3-Jan-2013)

Ref Expression
Assertion pm5.17 ( ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ↔ ¬ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 bicom ⊢ ( ( 𝜑 ↔ ¬ 𝜓 ) ↔ ( ¬ 𝜓 ↔ 𝜑 ) )
2 dfbi2 ⊢ ( ( ¬ 𝜓 ↔ 𝜑 ) ↔ ( ( ¬ 𝜓 → 𝜑 ) ∧ ( 𝜑 → ¬ 𝜓 ) ) )
3 orcom ⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜓 ∨ 𝜑 ) )
4 df-or ⊢ ( ( 𝜓 ∨ 𝜑 ) ↔ ( ¬ 𝜓 → 𝜑 ) )
5 3 4 bitr2i ⊢ ( ( ¬ 𝜓 → 𝜑 ) ↔ ( 𝜑 ∨ 𝜓 ) )
6 imnan ⊢ ( ( 𝜑 → ¬ 𝜓 ) ↔ ¬ ( 𝜑 ∧ 𝜓 ) )
7 5 6 anbi12i ⊢ ( ( ( ¬ 𝜓 → 𝜑 ) ∧ ( 𝜑 → ¬ 𝜓 ) ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜓 ) ) )
8 1 2 7 3bitrri ⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ↔ ¬ 𝜓 ) )