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Metamath Proof Explorer

Theorem 3bitrd

Description: Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999)

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

Proof

Step Hyp Ref Expression
1 3bitrd.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2 3bitrd.2 ⊢ ( 𝜑 → ( 𝜒 ↔ 𝜃 ) )
3 3bitrd.3 ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
4 1 2 bitrd ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
5 4 3 bitrd ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜏 ) )