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Theorem calemos 2019
 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 1429 . . . . 5 (𝜓 → ¬ 𝜒)
43con2i 557 . . . 4 (𝜒 → ¬ 𝜓)
5 calemos.maj . . . . 5 𝑥(𝜑𝜓)
65spi 1429 . . . 4 (𝜑𝜓)
74, 6nsyl 558 . . 3 (𝜒 → ¬ 𝜑)
87ancli 306 . 2 (𝜒 → (𝜒 ∧ ¬ 𝜑))
91, 8eximii 1493 1 𝑥(𝜒 ∧ ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110 This theorem is referenced by: (None)
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