Theorem barbari 2669

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 2663	. 2 ⊢ ∀𝑥(𝜒 → 𝜓)
5	1, 4	barbarilem 2668	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by:  celaront  2671  bamalip  2692