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Theorem calemes 2058
 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 1470 . . . 4 (𝜓 → ¬ 𝜒)
32con2i 590 . . 3 (𝜒 → ¬ 𝜓)
4 calemes.maj . . . 4 𝑥(𝜑𝜓)
54spi 1470 . . 3 (𝜑𝜓)
63, 5nsyl 591 . 2 (𝜒 → ¬ 𝜑)
76ax-gen 1379 1 𝑥(𝜒 → ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1283 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-in1 577  ax-in2 578  ax-gen 1379  ax-4 1441 This theorem is referenced by: (None)
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