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Metamath Proof Explorer

Theorem 3bitr4rd

Description: Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006)

Ref Expression
Hypotheses 3bitr4d.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3bitr4d.2 ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜓 ) )
3bitr4d.3 ⊢ ( 𝜑 → ( 𝜏 ↔ 𝜒 ) )
Assertion 3bitr4rd ( 𝜑 → ( 𝜏 ↔ 𝜃 ) )

Proof

Step Hyp Ref Expression
1 3bitr4d.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2 3bitr4d.2 ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜓 ) )
3 3bitr4d.3 ⊢ ( 𝜑 → ( 𝜏 ↔ 𝜒 ) )
4 3 1 bitr4d ⊢ ( 𝜑 → ( 𝜏 ↔ 𝜓 ) )
5 4 2 bitr4d ⊢ ( 𝜑 → ( 𝜏 ↔ 𝜃 ) )