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Metamath Proof Explorer

Theorem 3imtr3i

Description: A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994)

Ref Expression
Hypotheses 3imtr3.1 ⊢ ( 𝜑 → 𝜓 )
3imtr3.2 ⊢ ( 𝜑 ↔ 𝜒 )
3imtr3.3 ⊢ ( 𝜓 ↔ 𝜃 )
Assertion 3imtr3i ( 𝜒 → 𝜃 )

Proof

Step Hyp Ref Expression
1 3imtr3.1 ⊢ ( 𝜑 → 𝜓 )
2 3imtr3.2 ⊢ ( 𝜑 ↔ 𝜒 )
3 3imtr3.3 ⊢ ( 𝜓 ↔ 𝜃 )
4 2 1 sylbir ⊢ ( 𝜒 → 𝜓 )
5 4 3 sylib ⊢ ( 𝜒 → 𝜃 )