Theorem pm5.55

Description: Theorem *5.55 of WhiteheadRussell p. 125. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 20-Jan-2013)

	Ref	Expression
Assertion	pm5.55	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ↔ 𝜑 ) ∨ ( ( 𝜑 ∨ 𝜓 ) ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		biort	⊢ ( 𝜑 → ( 𝜑 ↔ ( 𝜑 ∨ 𝜓 ) ) )
2	1	bicomd	⊢ ( 𝜑 → ( ( 𝜑 ∨ 𝜓 ) ↔ 𝜑 ) )
3		biorf	⊢ ( ¬ 𝜑 → ( 𝜓 ↔ ( 𝜑 ∨ 𝜓 ) ) )
4	3	bicomd	⊢ ( ¬ 𝜑 → ( ( 𝜑 ∨ 𝜓 ) ↔ 𝜓 ) )
5	2 4	nsyl4	⊢ ( ¬ ( ( 𝜑 ∨ 𝜓 ) ↔ 𝜓 ) → ( ( 𝜑 ∨ 𝜓 ) ↔ 𝜑 ) )
6	5	con1i	⊢ ( ¬ ( ( 𝜑 ∨ 𝜓 ) ↔ 𝜑 ) → ( ( 𝜑 ∨ 𝜓 ) ↔ 𝜓 ) )
7	6	orri	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ↔ 𝜑 ) ∨ ( ( 𝜑 ∨ 𝜓 ) ↔ 𝜓 ) )