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Theorem camestros 2051
Description: "Camestros", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, no 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, AEO-2: PaM and SeM therefore SoP.) For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
camestros.maj 𝑥(𝜑𝜓)
camestros.min 𝑥(𝜒 → ¬ 𝜓)
camestros.e 𝑥𝜒
Assertion
Ref Expression
camestros 𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem camestros
StepHypRef Expression
1 camestros.e . 2 𝑥𝜒
2 camestros.min . . . . 5 𝑥(𝜒 → ¬ 𝜓)
32spi 1470 . . . 4 (𝜒 → ¬ 𝜓)
4 camestros.maj . . . . 5 𝑥(𝜑𝜓)
54spi 1470 . . . 4 (𝜑𝜓)
63, 5nsyl 591 . . 3 (𝜒 → ¬ 𝜑)
76ancli 316 . 2 (𝜒 → (𝜒 ∧ ¬ 𝜑))
81, 7eximii 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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