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Theorem calemes 2564
Description: "Calemes", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜓 is 𝜒, therefore no 𝜒 is 𝜑. (In Aristotelian notation, AEE-4: PaM and MeS therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
calemes.maj 𝑥(𝜑𝜓)
calemes.min 𝑥(𝜓 → ¬ 𝜒)
Assertion
Ref Expression
calemes 𝑥(𝜒 → ¬ 𝜑)

Proof of Theorem calemes
StepHypRef Expression
1 calemes.min . . . . 5 𝑥(𝜓 → ¬ 𝜒)
21spi 2039 . . . 4 (𝜓 → ¬ 𝜒)
32con2i 132 . . 3 (𝜒 → ¬ 𝜓)
4 calemes.maj . . . 4 𝑥(𝜑𝜓)
54spi 2039 . . 3 (𝜑𝜓)
63, 5nsyl 133 . 2 (𝜒 → ¬ 𝜑)
76ax-gen 1711 1 𝑥(𝜒 → ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-12 2031
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by: (None)
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