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Theorem baroco 2701
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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
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.maj . . 3 𝑥(𝜑𝜓)
2 con3 150 . . . . 5 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
32anim2d 605 . . . 4 ((𝜑𝜓) → ((𝜒 ∧ ¬ 𝜓) → (𝜒 ∧ ¬ 𝜑)))
43alimi 1906 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥((𝜒 ∧ ¬ 𝜓) → (𝜒 ∧ ¬ 𝜑)))
51, 4ax-mp 5 . 2 𝑥((𝜒 ∧ ¬ 𝜓) → (𝜒 ∧ ¬ 𝜑))
6 baroco.min . 2 𝑥(𝜒 ∧ ¬ 𝜓)
7 exim 1928 . 2 (∀𝑥((𝜒 ∧ ¬ 𝜓) → (𝜒 ∧ ¬ 𝜑)) → (∃𝑥(𝜒 ∧ ¬ 𝜓) → ∃𝑥(𝜒 ∧ ¬ 𝜑)))
85, 6, 7mp2 9 1 𝑥(𝜒 ∧ ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1650  wex 1874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875
This theorem is referenced by: (None)
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