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Theorem calemes 2319
 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 1753 . . . 4 (ψ → ¬ χ)
32con2i 112 . . 3 (χ → ¬ ψ)
4 calemes.maj . . . 4 x(φψ)
54spi 1753 . . 3 (φψ)
63, 5nsyl 113 . 2 (χ → ¬ φ)
76ax-gen 1546 1 x(χ → ¬ φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542 This theorem is referenced by: (None)
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