Theorem barbari 2752

Description: "Barbari", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is 𝜓. In Aristotelian notation, AAI-1: MaP and SaM therefore SiP. For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)

Hypotheses

Ref	Expression
barbari.maj	⊢ ∀𝑥(𝜑 → 𝜓)
barbari.min	⊢ ∀𝑥(𝜒 → 𝜑)
barbari.e	⊢ ∃𝑥𝜒

Assertion

Ref	Expression
barbari	⊢ ∃𝑥(𝜒 ∧ 𝜓)

Proof of Theorem barbari

Step	Hyp	Ref	Expression
1		barbari.e	. 2 ⊢ ∃𝑥𝜒
2		barbari.maj	. . 3 ⊢ ∀𝑥(𝜑 → 𝜓)
3		barbari.min	. . 3 ⊢ ∀𝑥(𝜒 → 𝜑)
4	2, 3	barbara 2746	. 2 ⊢ ∀𝑥(𝜒 → 𝜓)
5	1, 4	barbarilem 2751	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1654  ∃wex 1878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879

This theorem is referenced by:  celaront  2754  bamalip  2786