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Theorem calemos 2096
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 1501 . . . . 5 (𝜓 → ¬ 𝜒)
43con2i 601 . . . 4 (𝜒 → ¬ 𝜓)
5 calemos.maj . . . . 5 𝑥(𝜑𝜓)
65spi 1501 . . . 4 (𝜑𝜓)
74, 6nsyl 602 . . 3 (𝜒 → ¬ 𝜑)
87ancli 321 . 2 (𝜒 → (𝜒 ∧ ¬ 𝜑))
91, 8eximii 1566 1 𝑥(𝜒 ∧ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wal 1314  wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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