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Metamath Proof Explorer

Theorem biimtrrid

Description: A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993)

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

Proof

Step Hyp Ref Expression
1 biimtrrid.1 ⊢ ( 𝜓 ↔ 𝜑 )
2 biimtrrid.2 ⊢ ( 𝜒 → ( 𝜓 → 𝜃 ) )
3 1 biimpri ⊢ ( 𝜑 → 𝜓 )
4 3 2 syl5 ⊢ ( 𝜒 → ( 𝜑 → 𝜃 ) )