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 ) )