Theorem pm5.55 867

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 866	. . . . 5 ⊢ (φ → (φ ↔ (φ ∨ ψ)))
2	1	bicomd 192	. . . 4 ⊢ (φ → ((φ ∨ ψ) ↔ φ))
3		biorf 394	. . . . 5 ⊢ (¬ φ → (ψ ↔ (φ ∨ ψ)))
4	3	bicomd 192	. . . 4 ⊢ (¬ φ → ((φ ∨ ψ) ↔ ψ))
5	2, 4	nsyl4 134	. . 3 ⊢ (¬ ((φ ∨ ψ) ↔ ψ) → ((φ ∨ ψ) ↔ φ))
6	5	con1i 121	. 2 ⊢ (¬ ((φ ∨ ψ) ↔ φ) → ((φ ∨ ψ) ↔ ψ))
7	6	orri 365	1 ⊢ (((φ ∨ ψ) ↔ φ) ∨ ((φ ∨ ψ) ↔ ψ))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357

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 177  df-or 359

This theorem is referenced by: (None)