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 ⊢ φ → ψ ↔ ψ ↔ φ ∨ ψ

Proof

Step Hyp Ref Expression
1 olc ⊢ ψ → φ ∨ ψ
2 pm2.621 ⊢ φ → ψ → φ ∨ ψ → ψ
3 1 2 impbid2 ⊢ φ → ψ → ψ ↔ φ ∨ ψ
4 orc ⊢ φ → φ ∨ ψ
5 biimpr ⊢ ψ ↔ φ ∨ ψ → φ ∨ ψ → ψ
6 4 5 syl5 ⊢ ψ ↔ φ ∨ ψ → φ → ψ
7 3 6 impbii ⊢ φ → ψ ↔ ψ ↔ φ ∨ ψ