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Metamath Proof Explorer

Theorem 3imtr4i

Description: A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 3-Jan-1993)

Ref Expression
Hypotheses 3imtr4.1 
|- ( ph -> ps )
3imtr4.2 
|- ( ch <-> ph )
3imtr4.3 
|- ( th <-> ps )
Assertion 3imtr4i 
|- ( ch -> th )

Proof

Step Hyp Ref Expression
1 3imtr4.1 
 |-  ( ph -> ps )
2 3imtr4.2 
 |-  ( ch <-> ph )
3 3imtr4.3 
 |-  ( th <-> ps )
4 2 1 sylbi 
 |-  ( ch -> ps )
5 4 3 sylibr 
 |-  ( ch -> th )