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Theorem disamis 2767
 Description: "Disamis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. 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.)
Hypotheses
Ref Expression
disamis.maj 𝑥(𝜑𝜓)
disamis.min 𝑥(𝜑𝜒)
Assertion
Ref Expression
disamis 𝑥(𝜒𝜓)

Proof of Theorem disamis
StepHypRef Expression
1 disamis.min . . 3 𝑥(𝜑𝜒)
2 disamis.maj . . 3 𝑥(𝜑𝜓)
31, 2datisi 2766 . 2 𝑥(𝜓𝜒)
4 exancom 1854 . 2 (∃𝑥(𝜓𝜒) ↔ ∃𝑥(𝜒𝜓))
53, 4mpbi 231 1 𝑥(𝜒𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1528  ∃wex 1773 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774 This theorem is referenced by:  bocardo  2769
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