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Metamath Proof Explorer

Theorem cesaro

Description: "Cesaro", one of the syllogisms of Aristotelian logic. No ph is ps , all ch is ps , and ch exist, therefore some ch is not ph . In Aristotelian notation, EAO-2: PeM and SaM therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

Ref Expression
Hypotheses cesaro.maj 
|- A. x ( ph -> -. ps )
cesaro.min 
|- A. x ( ch -> ps )
cesaro.e 
|- E. x ch
Assertion cesaro 
|- E. x ( ch /\ -. ph )

Proof

Step Hyp Ref Expression
1 cesaro.maj 
 |-  A. x ( ph -> -. ps )
2 cesaro.min 
 |-  A. x ( ch -> ps )
3 cesaro.e 
 |-  E. x ch
4 1 2 cesare 
 |-  A. x ( ch -> -. ph )
5 3 4 barbarilem 
 |-  E. x ( ch /\ -. ph )