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Theorem datisi 2766
 Description: "Datisi", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. 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.)
Hypotheses
Ref Expression
datisi.maj 𝑥(𝜑𝜓)
datisi.min 𝑥(𝜑𝜒)
Assertion
Ref Expression
datisi 𝑥(𝜒𝜓)

Proof of Theorem datisi
StepHypRef Expression
1 datisi.maj . 2 𝑥(𝜑𝜓)
2 datisi.min . . 3 𝑥(𝜑𝜒)
3 exancom 1854 . . 3 (∃𝑥(𝜑𝜒) ↔ ∃𝑥(𝜒𝜑))
42, 3mpbi 231 . 2 𝑥(𝜒𝜑)
51, 4darii 2751 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:  disamis  2767  ferison  2768
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