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Theorem baroco 2310
 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 . . . . . 6 x(φψ)
32spi 1753 . . . . 5 (φψ)
43con3i 127 . . . 4 ψ → ¬ φ)
54anim2i 552 . . 3 ((χ ¬ ψ) → (χ ¬ φ))
65eximi 1576 . 2 (x(χ ¬ ψ) → x(χ ¬ φ))
71, 6ax-mp 5 1 x(χ ¬ φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-an 360  df-ex 1542 This theorem is referenced by: (None)
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