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Metamath Proof Explorer

Theorem baroco

Description: "Baroco", one of the syllogisms of Aristotelian logic. All ph is ps , and some ch is not ps , therefore some ch is not ph . In Aristotelian notation, AOO-2: PaM and SoM therefore SoP. For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative". (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step Hyp Ref Expression
1 baroco.maj 
 |-  A. x ( ph -> ps )
2 baroco.min 
 |-  E. x ( ch /\ -. ps )
3 con3 
 |-  ( ( ph -> ps ) -> ( -. ps -> -. ph ) )
4 3 anim2d 
 |-  ( ( ph -> ps ) -> ( ( ch /\ -. ps ) -> ( ch /\ -. ph ) ) )
5 4 alimi 
 |-  ( A. x ( ph -> ps ) -> A. x ( ( ch /\ -. ps ) -> ( ch /\ -. ph ) ) )
6 1 5 ax-mp 
 |-  A. x ( ( ch /\ -. ps ) -> ( ch /\ -. ph ) )
7 exim 
 |-  ( A. x ( ( ch /\ -. ps ) -> ( ch /\ -. ph ) ) -> ( E. x ( ch /\ -. ps ) -> E. x ( ch /\ -. ph ) ) )
8 6 2 7 mp2 
 |-  E. x ( ch /\ -. ph )