Theorem oibabs 851

Description: Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)

Assertion

Ref	Expression
oibabs	⊢ (((φ ∨ ψ) → (φ ↔ ψ)) ↔ (φ ↔ ψ))

Proof of Theorem oibabs

Step	Hyp	Ref	Expression
1		ioran 476	. . . 4 ⊢ (¬ (φ ∨ ψ) ↔ (¬ φ ∧ ¬ ψ))
2		pm5.21 831	. . . 4 ⊢ ((¬ φ ∧ ¬ ψ) → (φ ↔ ψ))
3	1, 2	sylbi 187	. . 3 ⊢ (¬ (φ ∨ ψ) → (φ ↔ ψ))
4		id 19	. . 3 ⊢ ((φ ↔ ψ) → (φ ↔ ψ))
5	3, 4	ja 153	. 2 ⊢ (((φ ∨ ψ) → (φ ↔ ψ)) → (φ ↔ ψ))
6		ax-1 6	. 2 ⊢ ((φ ↔ ψ) → ((φ ∨ ψ) → (φ ↔ ψ)))
7	5, 6	impbii 180	1 ⊢ (((φ ∨ ψ) → (φ ↔ ψ)) ↔ (φ ↔ ψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

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  df-an 360

This theorem is referenced by: (None)