Theorem pm4.83 895

Description: Theorem *4.83 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm4.83	⊢ (((φ → ψ) ∧ (¬ φ → ψ)) ↔ ψ)

Proof of Theorem pm4.83

Step	Hyp	Ref	Expression
1		exmid 404	. . 3 ⊢ (φ ∨ ¬ φ)
2	1	a1bi 327	. 2 ⊢ (ψ ↔ ((φ ∨ ¬ φ) → ψ))
3		jaob 758	. 2 ⊢ (((φ ∨ ¬ φ) → ψ) ↔ ((φ → ψ) ∧ (¬ φ → ψ)))
4	2, 3	bitr2i 241	1 ⊢ (((φ → ψ) ∧ (¬ φ → ψ)) ↔ 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)