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Theorem calemos 2613
Description: "Calemos", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓 (PaM), no 𝜓 is 𝜒 (MeS), and 𝜒 exist, therefore some 𝜒 is not 𝜑 (SoP). (In Aristotelian notation, AEO-4: PaM and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
calemos.maj 𝑥(𝜑𝜓)
calemos.min 𝑥(𝜓 → ¬ 𝜒)
calemos.e 𝑥𝜒
Assertion
Ref Expression
calemos 𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem calemos
StepHypRef Expression
1 calemos.e . 2 𝑥𝜒
2 calemos.min . . . . . 6 𝑥(𝜓 → ¬ 𝜒)
32spi 2092 . . . . 5 (𝜓 → ¬ 𝜒)
43con2i 134 . . . 4 (𝜒 → ¬ 𝜓)
5 calemos.maj . . . . 5 𝑥(𝜑𝜓)
65spi 2092 . . . 4 (𝜑𝜓)
74, 6nsyl 135 . . 3 (𝜒 → ¬ 𝜑)
87ancli 573 . 2 (𝜒 → (𝜒 ∧ ¬ 𝜑))
91, 8eximii 1804 1 𝑥(𝜒 ∧ ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by: (None)
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