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 ( 𝜑𝜓 )
syl7bi.2 ( 𝜒 → ( 𝜃 → ( 𝜓𝜏 ) ) )
Assertion syl7bi ( 𝜒 → ( 𝜃 → ( 𝜑𝜏 ) ) )

Proof

Step Hyp Ref Expression
1 syl7bi.1 ( 𝜑𝜓 )
2 syl7bi.2 ( 𝜒 → ( 𝜃 → ( 𝜓𝜏 ) ) )
3 1 biimpi ( 𝜑𝜓 )
4 3 2 syl7 ( 𝜒 → ( 𝜃 → ( 𝜑𝜏 ) ) )