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Metamath Proof Explorer

Theorem datisi

Description: "Datisi", one of the syllogisms of Aristotelian logic. All ph is ps , and some ph is ch , therefore some ch is ps . In Aristotelian notation, AII-3: MaP and MiS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		datisi.maj	\|- A. x ( ph -> ps )
2		datisi.min	\|- E. x ( ph /\ ch )
3		exancom	\|- ( E. x ( ph /\ ch ) <-> E. x ( ch /\ ph ) )
4	2 3	mpbi	\|- E. x ( ch /\ ph )
5	1 4	darii	\|- E. x ( ch /\ ps )