Metamath Proof Explorer


Theorem sylnibr

Description: A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013)

Ref Expression
Hypotheses sylnibr.1
|- ( ph -> -. ps )
sylnibr.2
|- ( ch <-> ps )
Assertion sylnibr
|- ( ph -> -. ch )

Proof

Step Hyp Ref Expression
1 sylnibr.1
 |-  ( ph -> -. ps )
2 sylnibr.2
 |-  ( ch <-> ps )
3 2 bicomi
 |-  ( ps <-> ch )
4 1 3 sylnib
 |-  ( ph -> -. ch )