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Theorem festino 2570
Description: "Festino", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜓, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)
Hypotheses
Ref Expression
festino.maj 𝑥(𝜑 → ¬ 𝜓)
festino.min 𝑥(𝜒𝜓)
Assertion
Ref Expression
festino 𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem festino
StepHypRef Expression
1 festino.min . 2 𝑥(𝜒𝜓)
2 festino.maj . . . . 5 𝑥(𝜑 → ¬ 𝜓)
32spi 2052 . . . 4 (𝜑 → ¬ 𝜓)
43con2i 134 . . 3 (𝜓 → ¬ 𝜑)
54anim2i 592 . 2 ((𝜒𝜓) → (𝜒 ∧ ¬ 𝜑))
61, 5eximii 1761 1 𝑥(𝜒 ∧ ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
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