Metamath Proof Explorer


Theorem pm2.86

Description: Converse of Axiom ax-2 . Theorem *2.86 of WhiteheadRussell p. 108. (Contributed by NM, 25-Apr-1994) (Proof shortened by Wolf Lammen, 3-Apr-2013)

Ref Expression
Assertion pm2.86 ( ( ( 𝜑𝜓 ) → ( 𝜑𝜒 ) ) → ( 𝜑 → ( 𝜓𝜒 ) ) )

Proof

Step Hyp Ref Expression
1 id ( ( ( 𝜑𝜓 ) → ( 𝜑𝜒 ) ) → ( ( 𝜑𝜓 ) → ( 𝜑𝜒 ) ) )
2 1 pm2.86d ( ( ( 𝜑𝜓 ) → ( 𝜑𝜒 ) ) → ( 𝜑 → ( 𝜓𝜒 ) ) )