Metamath Proof Explorer

Theorem bitr

Description: Theorem *4.22 of WhiteheadRussell p. 117. bitri in closed form. (Contributed by NM, 3-Jan-2005)

Ref Expression
Assertion bitr ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜓 ↔ 𝜒 ) ) → ( 𝜑 ↔ 𝜒 ) )

Proof

Step Hyp Ref Expression
1 bibi1 ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜑 ↔ 𝜒 ) ↔ ( 𝜓 ↔ 𝜒 ) ) )
2 1 biimpar ⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜓 ↔ 𝜒 ) ) → ( 𝜑 ↔ 𝜒 ) )