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Theorem calemos 2736
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 2211 . . . . 5 (𝜓 → ¬ 𝜒)
43con2i 136 . . . 4 (𝜒 → ¬ 𝜓)
5 calemos.maj . . . . 5 𝑥(𝜑𝜓)
65spi 2211 . . . 4 (𝜑𝜓)
74, 6nsyl 137 . . 3 (𝜒 → ¬ 𝜑)
87ancli 539 . 2 (𝜒 → (𝜒 ∧ ¬ 𝜑))
91, 8eximii 1915 1 𝑥(𝜒 ∧ ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1632  wex 1855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-12 2206
This theorem depends on definitions:  df-bi 198  df-an 384  df-ex 1856
This theorem is referenced by: (None)
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