Theorem datisi 2680

Description: "Datisi", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. In Aristotelian notation, AII-3: MaP and MiS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)

Hypotheses

Ref	Expression
datisi.maj	⊢ ∀𝑥(𝜑 → 𝜓)
datisi.min	⊢ ∃𝑥(𝜑 ∧ 𝜒)

Assertion

Ref	Expression
datisi	⊢ ∃𝑥(𝜒 ∧ 𝜓)

Proof of Theorem datisi

Step	Hyp	Ref	Expression
1		datisi.maj	. 2 ⊢ ∀𝑥(𝜑 → 𝜓)
2		datisi.min	. . 3 ⊢ ∃𝑥(𝜑 ∧ 𝜒)
3		exancom 1861	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜒) ↔ ∃𝑥(𝜒 ∧ 𝜑))
4	2, 3	mpbi 230	. 2 ⊢ ∃𝑥(𝜒 ∧ 𝜑)
5	1, 4	darii 2665	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:  disamis  2681  ferison  2682