Metamath Proof Explorer


Theorem disamis

Description: "Disamis", one of the syllogisms of Aristotelian logic. Some ph is ps , and all ph is ch , therefore some ch is ps . In Aristotelian notation, IAI-3: MiP and MaS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

Ref Expression
Hypotheses disamis.maj 𝑥 ( 𝜑𝜓 )
disamis.min 𝑥 ( 𝜑𝜒 )
Assertion disamis 𝑥 ( 𝜒𝜓 )

Proof

Step Hyp Ref Expression
1 disamis.maj 𝑥 ( 𝜑𝜓 )
2 disamis.min 𝑥 ( 𝜑𝜒 )
3 2 1 datisi 𝑥 ( 𝜓𝜒 )
4 exancom ( ∃ 𝑥 ( 𝜓𝜒 ) ↔ ∃ 𝑥 ( 𝜒𝜓 ) )
5 3 4 mpbi 𝑥 ( 𝜒𝜓 )