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Mirrors > Home > ILE Home > Th. List > ax-2 | GIF version |
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 pm5.41 250. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
ax-2 | ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | wps | . . . 4 wff 𝜓 | |
3 | wch | . . . 4 wff 𝜒 | |
4 | 2, 3 | wi 4 | . . 3 wff (𝜓 → 𝜒) |
5 | 1, 4 | wi 4 | . 2 wff (𝜑 → (𝜓 → 𝜒)) |
6 | 1, 2 | wi 4 | . . 3 wff (𝜑 → 𝜓) |
7 | 1, 3 | wi 4 | . . 3 wff (𝜑 → 𝜒) |
8 | 6, 7 | wi 4 | . 2 wff ((𝜑 → 𝜓) → (𝜑 → 𝜒)) |
9 | 5, 8 | wi 4 | 1 wff ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) |
Colors of variables: wff set class |
This axiom is referenced by: a2i 11 idALT 20 a2d 26 imdi 249 sbi1v 1863 ralim 2491 |
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