Theorem pm5.55 950

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

Assertion

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

Proof of Theorem pm5.55

Step	Hyp	Ref	Expression
1		biort 935	. . . 4 ⊢ (𝜑 → (𝜑 ↔ (𝜑 ∨ 𝜓)))
2	1	bicomd 223	. . 3 ⊢ (𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜑))
3		biorf 936	. . . 4 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))
4	3	bicomd 223	. . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓))
5	2, 4	nsyl5 159	. 2 ⊢ (¬ ((𝜑 ∨ 𝜓) ↔ 𝜑) → ((𝜑 ∨ 𝜓) ↔ 𝜓))
6	5	orri 862	1 ⊢ (((𝜑 ∨ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∨ 𝜓) ↔ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∨ wo 847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-or 848

This theorem is referenced by: (None)