This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 3bitr2i

Description: A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006)

Ref Expression
Hypotheses 3bitr2i.1 ⊢ ( 𝜑 ↔ 𝜓 )
3bitr2i.2 ⊢ ( 𝜒 ↔ 𝜓 )
3bitr2i.3 ⊢ ( 𝜒 ↔ 𝜃 )
Assertion 3bitr2i ( 𝜑 ↔ 𝜃 )

Proof

Step Hyp Ref Expression
1 3bitr2i.1 ⊢ ( 𝜑 ↔ 𝜓 )
2 3bitr2i.2 ⊢ ( 𝜒 ↔ 𝜓 )
3 3bitr2i.3 ⊢ ( 𝜒 ↔ 𝜃 )
4 1 2 bitr4i ⊢ ( 𝜑 ↔ 𝜒 )
5 4 3 bitri ⊢ ( 𝜑 ↔ 𝜃 )