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Theorem calemos 2062
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 1470 . . . . 5 (𝜓 → ¬ 𝜒)
43con2i 590 . . . 4 (𝜒 → ¬ 𝜓)
5 calemos.maj . . . . 5 𝑥(𝜑𝜓)
65spi 1470 . . . 4 (𝜑𝜓)
74, 6nsyl 591 . . 3 (𝜒 → ¬ 𝜑)
87ancli 316 . 2 (𝜒 → (𝜒 ∧ ¬ 𝜑))
91, 8eximii 1534 1 𝑥(𝜒 ∧ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wal 1283  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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