# Metamath Proof Explorer

## Theorem anass

Description: Associative law for conjunction. Theorem *4.32 of WhiteheadRussell p. 118. (Contributed by NM, 21-Jun-1993) (Proof shortened by Wolf Lammen, 24-Nov-2012)

Ref Expression
Assertion anass ${⊢}\left(\left({\phi }\wedge {\psi }\right)\wedge {\chi }\right)↔\left({\phi }\wedge \left({\psi }\wedge {\chi }\right)\right)$

### Proof

Step Hyp Ref Expression
1 id ${⊢}\left({\phi }\wedge \left({\psi }\wedge {\chi }\right)\right)\to \left({\phi }\wedge \left({\psi }\wedge {\chi }\right)\right)$
2 1 anassrs ${⊢}\left(\left({\phi }\wedge {\psi }\right)\wedge {\chi }\right)\to \left({\phi }\wedge \left({\psi }\wedge {\chi }\right)\right)$
3 id ${⊢}\left(\left({\phi }\wedge {\psi }\right)\wedge {\chi }\right)\to \left(\left({\phi }\wedge {\psi }\right)\wedge {\chi }\right)$
4 3 anasss ${⊢}\left({\phi }\wedge \left({\psi }\wedge {\chi }\right)\right)\to \left(\left({\phi }\wedge {\psi }\right)\wedge {\chi }\right)$
5 2 4 impbii ${⊢}\left(\left({\phi }\wedge {\psi }\right)\wedge {\chi }\right)↔\left({\phi }\wedge \left({\psi }\wedge {\chi }\right)\right)$