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Metamath Proof Explorer

Theorem 3bitr3g

Description: More general version of 3bitr3i . Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995)

Ref Expression
Hypotheses 3bitr3g.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3bitr3g.2 ⊢ ( 𝜓 ↔ 𝜃 )
3bitr3g.3 ⊢ ( 𝜒 ↔ 𝜏 )
Assertion 3bitr3g ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )

Proof

Step Hyp Ref Expression
1 3bitr3g.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2 3bitr3g.2 ⊢ ( 𝜓 ↔ 𝜃 )
3 3bitr3g.3 ⊢ ( 𝜒 ↔ 𝜏 )
4 2 1 bitr3id ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜒 ) )
5 4 3 bitrdi ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )