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Theorem baroco 2570
Description: "Baroco", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is not 𝜓, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, AOO-2: PaM and SoM therefore SoP.) For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative." (Contributed by David A. Wheeler, 28-Aug-2016.)
Hypotheses
Ref Expression
baroco.maj 𝑥(𝜑𝜓)
baroco.min 𝑥(𝜒 ∧ ¬ 𝜓)
Assertion
Ref Expression
baroco 𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem baroco
StepHypRef Expression
1 baroco.min . 2 𝑥(𝜒 ∧ ¬ 𝜓)
2 baroco.maj . . . . 5 𝑥(𝜑𝜓)
32spi 2052 . . . 4 (𝜑𝜓)
43con3i 150 . . 3 𝜓 → ¬ 𝜑)
54anim2i 592 . 2 ((𝜒 ∧ ¬ 𝜓) → (𝜒 ∧ ¬ 𝜑))
61, 5eximii 1762 1 𝑥(𝜒 ∧ ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1479  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703
This theorem is referenced by: (None)
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