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Metamath Proof Explorer

Theorem 3imtr4i

Description: A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 3-Jan-1993)

Ref Expression
Hypotheses 3imtr4.1 ⊢ ( 𝜑 → 𝜓 )
3imtr4.2 ⊢ ( 𝜒 ↔ 𝜑 )
3imtr4.3 ⊢ ( 𝜃 ↔ 𝜓 )
Assertion 3imtr4i ( 𝜒 → 𝜃 )

Proof

Step Hyp Ref Expression
1 3imtr4.1 ⊢ ( 𝜑 → 𝜓 )
2 3imtr4.2 ⊢ ( 𝜒 ↔ 𝜑 )
3 3imtr4.3 ⊢ ( 𝜃 ↔ 𝜓 )
4 2 1 sylbi ⊢ ( 𝜒 → 𝜓 )
5 4 3 sylibr ⊢ ( 𝜒 → 𝜃 )