Theorem barbari 2044

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.) (Revised by David A. Wheeler, 30-Aug-2016.)

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	. . . . 5 ⊢ ∀𝑥(𝜑 → 𝜓)
3		barbari.min	. . . . 5 ⊢ ∀𝑥(𝜒 → 𝜑)
4	2, 3	barbara 2040	. . . 4 ⊢ ∀𝑥(𝜒 → 𝜓)
5	4	spi 1470	. . 3 ⊢ (𝜒 → 𝜓)
6	5	ancli 316	. 2 ⊢ (𝜒 → (𝜒 ∧ 𝜓))
7	1, 6	eximii 1534	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1283  ∃wex 1422

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  celaront  2045