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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 pm5.41 373. (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  199  imdi  372  sbi1  2275  difin0ss  3780  rp-fakeimass  36396
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