# Metamath Proof Explorer

## Axiom ax-2

Description: AxiomFrege. 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 asFrege 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 . (Contributed by NM, 30-Sep-1992)

Ref Expression
Assertion ax-2 ${⊢}\left({\phi }\to \left({\psi }\to {\chi }\right)\right)\to \left(\left({\phi }\to {\psi }\right)\to \left({\phi }\to {\chi }\right)\right)$

### Detailed syntax breakdown

Step Hyp Ref Expression
0 wph ${wff}{\phi }$
1 wps ${wff}{\psi }$
2 wch ${wff}{\chi }$
3 1 2 wi ${wff}\left({\psi }\to {\chi }\right)$
4 0 3 wi ${wff}\left({\phi }\to \left({\psi }\to {\chi }\right)\right)$
5 0 1 wi ${wff}\left({\phi }\to {\psi }\right)$
6 0 2 wi ${wff}\left({\phi }\to {\chi }\right)$
7 5 6 wi ${wff}\left(\left({\phi }\to {\psi }\right)\to \left({\phi }\to {\chi }\right)\right)$
8 4 7 wi ${wff}\left(\left({\phi }\to \left({\psi }\to {\chi }\right)\right)\to \left(\left({\phi }\to {\psi }\right)\to \left({\phi }\to {\chi }\right)\right)\right)$