# Metamath Proof Explorer

## Theorem sylbbr

Description: A mixed syllogism inference from two biconditionals.

Note on the various syllogism-like statements in set.mm. The hypothetical syllogism syl infers an implication from two implications (and there are 3syl and 4syl for chaining more inferences). There are four inferences inferring an implication from one implication and one biconditional: sylbi , sylib , sylbir , sylibr ; four inferences inferring an implication from two biconditionals: sylbb , sylbbr , sylbb1 , sylbb2 ; four inferences inferring a biconditional from two biconditionals: bitri , bitr2i , bitr3i , bitr4i (and more for chaining more biconditionals). There are also closed forms and deduction versions of these, like, among many others, syld , syl5 , syl6 , mpbid , bitrd , syl5bb , syl6bb and variants. (Contributed by BJ, 21-Apr-2019)

Ref Expression
Hypotheses sylbbr.1 ( 𝜑𝜓 )
sylbbr.2 ( 𝜓𝜒 )
Assertion sylbbr ( 𝜒𝜑 )

### Proof

Step Hyp Ref Expression
1 sylbbr.1 ( 𝜑𝜓 )
2 sylbbr.2 ( 𝜓𝜒 )
3 2 biimpri ( 𝜒𝜓 )
4 3 1 sylibr ( 𝜒𝜑 )