# Metamath Proof Explorer

## Theorem pm4.72

Description: Implication in terms of biconditional and disjunction. Theorem *4.72 of WhiteheadRussell p. 121. (Contributed by NM, 30-Aug-1993) (Proof shortened by Wolf Lammen, 30-Jan-2013)

Ref Expression
Assertion pm4.72 ${⊢}\left({\phi }\to {\psi }\right)↔\left({\psi }↔\left({\phi }\vee {\psi }\right)\right)$

### Proof

Step Hyp Ref Expression
1 olc ${⊢}{\psi }\to \left({\phi }\vee {\psi }\right)$
2 pm2.621 ${⊢}\left({\phi }\to {\psi }\right)\to \left(\left({\phi }\vee {\psi }\right)\to {\psi }\right)$
3 1 2 impbid2 ${⊢}\left({\phi }\to {\psi }\right)\to \left({\psi }↔\left({\phi }\vee {\psi }\right)\right)$
4 orc ${⊢}{\phi }\to \left({\phi }\vee {\psi }\right)$
5 biimpr ${⊢}\left({\psi }↔\left({\phi }\vee {\psi }\right)\right)\to \left(\left({\phi }\vee {\psi }\right)\to {\psi }\right)$
6 4 5 syl5 ${⊢}\left({\psi }↔\left({\phi }\vee {\psi }\right)\right)\to \left({\phi }\to {\psi }\right)$
7 3 6 impbii ${⊢}\left({\phi }\to {\psi }\right)↔\left({\psi }↔\left({\phi }\vee {\psi }\right)\right)$