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Metamath Proof Explorer

Theorem syl7bi

Description: A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 14-May-1993)

Ref Expression
Hypotheses syl7bi.1 ⊢ ( 𝜑 ↔ 𝜓 )
syl7bi.2 ⊢ ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) )
Assertion syl7bi ( 𝜒 → ( 𝜃 → ( 𝜑 → 𝜏 ) ) )

Proof

Step Hyp Ref Expression
1 syl7bi.1 ⊢ ( 𝜑 ↔ 𝜓 )
2 syl7bi.2 ⊢ ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) )
3 1 biimpi ⊢ ( 𝜑 → 𝜓 )
4 3 2 syl7 ⊢ ( 𝜒 → ( 𝜃 → ( 𝜑 → 𝜏 ) ) )