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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 
|- ( ph -> ps )
3imtr3.2 
|- ( ph <-> ch )
3imtr3.3 
|- ( ps <-> th )
Assertion 3imtr3i 
|- ( ch -> th )

Proof

Step Hyp Ref Expression
1 3imtr3.1 
 |-  ( ph -> ps )
2 3imtr3.2 
 |-  ( ph <-> ch )
3 3imtr3.3 
 |-  ( ps <-> th )
4 2 1 sylbir 
 |-  ( ch -> ps )
5 4 3 sylib 
 |-  ( ch -> th )