Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-2 GIF version

Axiom ax-2 6
 Description: Axiom Frege. Axiom A2 of [Margaris] p. 49. This axiom "distributes" an antecedent over two consequents. This axiom was part of Frege's original system and is known as Frege in the literature. It is also proved as Theorem *2.77 of [WhiteheadRussell] p. 108. The other direction of this axiom also turns out to be true, as demonstrated by pm5.41 240. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ax-2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Detailed syntax breakdown of Axiom ax-2
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 wps . . . 4 wff 𝜓
3 wch . . . 4 wff 𝜒
42, 3wi 4 . . 3 wff (𝜓𝜒)
51, 4wi 4 . 2 wff (𝜑 → (𝜓𝜒))
61, 2wi 4 . . 3 wff (𝜑𝜓)
71, 3wi 4 . . 3 wff (𝜑𝜒)
86, 7wi 4 . 2 wff ((𝜑𝜓) → (𝜑𝜒))
95, 8wi 4 1 wff ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
 Colors of variables: wff set class This axiom is referenced by:  a2i  11  idALT  20  a2d  23  imdi  239  sbi1v  1771  ralim  2380
 Copyright terms: Public domain W3C validator