Theorem barbari 2121

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 2117	. . . 4 ⊢ ∀𝑥(𝜒 → 𝜓)
5	4	spi 1529	. . 3 ⊢ (𝜒 → 𝜓)
6	5	ancli 321	. 2 ⊢ (𝜒 → (𝜒 ∧ 𝜓))
7	1, 6	eximii 1595	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1346  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  celaront  2122