MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  darapti Structured version   Visualization version   GIF version

Theorem darapti 2769
Description: "Darapti", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is 𝜓. In Aristotelian notation, AAI-3: MaP and MaS therefore SiP. For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
Hypotheses
Ref Expression
darapti.maj 𝑥(𝜑𝜓)
darapti.min 𝑥(𝜑𝜒)
darapti.e 𝑥𝜑
Assertion
Ref Expression
darapti 𝑥(𝜒𝜓)

Proof of Theorem darapti
StepHypRef Expression
1 darapti.min . . . 4 𝑥(𝜑𝜒)
2 darapti.maj . . . 4 𝑥(𝜑𝜓)
3 id 22 . . . . 5 (((𝜑𝜒) ∧ (𝜑𝜓)) → ((𝜑𝜒) ∧ (𝜑𝜓)))
43alanimi 1818 . . . 4 ((∀𝑥(𝜑𝜒) ∧ ∀𝑥(𝜑𝜓)) → ∀𝑥((𝜑𝜒) ∧ (𝜑𝜓)))
51, 2, 4mp2an 691 . . 3 𝑥((𝜑𝜒) ∧ (𝜑𝜓))
6 pm3.43 477 . . . 4 (((𝜑𝜒) ∧ (𝜑𝜓)) → (𝜑 → (𝜒𝜓)))
76alimi 1813 . . 3 (∀𝑥((𝜑𝜒) ∧ (𝜑𝜓)) → ∀𝑥(𝜑 → (𝜒𝜓)))
85, 7ax-mp 5 . 2 𝑥(𝜑 → (𝜒𝜓))
9 darapti.e . 2 𝑥𝜑
10 exim 1835 . 2 (∀𝑥(𝜑 → (𝜒𝜓)) → (∃𝑥𝜑 → ∃𝑥(𝜒𝜓)))
118, 9, 10mp2 9 1 𝑥(𝜒𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  felapton  2771
  Copyright terms: Public domain W3C validator