Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-2 Structured version   Visualization version   GIF version

Axiom ax-2 7
 Description: Axiom Frege. Axiom A2 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It "distributes" an antecedent over two consequents. This axiom was part of Frege's original system and is known as Frege in the literature; see Proposition 2 of [Frege1879] p. 26. 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 pm5.41 395. (Contributed by NM, 30-Sep-1992.)
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 setvar class This axiom is referenced by:  a2i  14  idALT  23  a2d  29  dfbi1ALT  217  imdi  394  sbi1  2076  sbi1vOLD  2317  sbi1OLD  2519  difin0ss  4267  bj-sblem1  34562  bj-sblem2  34563  rp-fakeimass  40593  adh-minim  43960  adh-minimp  43972
 Copyright terms: Public domain W3C validator