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Metamath Proof Explorer

Theorem 3bitr4g

Description: More general version of 3bitr4i . Useful for converting definitions in a formula. (Contributed by NM, 11-May-1993)

Ref Expression
Hypotheses 3bitr4g.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3bitr4g.2 ⊢ ( 𝜃 ↔ 𝜓 )
3bitr4g.3 ⊢ ( 𝜏 ↔ 𝜒 )
Assertion 3bitr4g ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )

Proof

Step Hyp Ref Expression
1 3bitr4g.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2 3bitr4g.2 ⊢ ( 𝜃 ↔ 𝜓 )
3 3bitr4g.3 ⊢ ( 𝜏 ↔ 𝜒 )
4 2 1 bitrid ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜒 ) )
5 4 3 bitr4di ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )