Theorem dimatis 2714

Description: "Dimatis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜓 is 𝜒, therefore some 𝜒 is 𝜑. In Aristotelian notation, IAI-4: PiM and MaS therefore SiP. For example, "Some pets are rabbits", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 2691 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)

Hypotheses

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

Assertion

Ref	Expression
dimatis	⊢ ∃𝑥(𝜒 ∧ 𝜑)

Proof of Theorem dimatis

Step	Hyp	Ref	Expression
1		dimatis.min	. . 3 ⊢ ∀𝑥(𝜓 → 𝜒)
2		dimatis.maj	. . 3 ⊢ ∃𝑥(𝜑 ∧ 𝜓)
3	1, 2	darii 2691	. 2 ⊢ ∃𝑥(𝜑 ∧ 𝜒)
4		exancom 1881	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜒) ↔ ∃𝑥(𝜒 ∧ 𝜑))
5	3, 4	mpbi 232	1 ⊢ ∃𝑥(𝜒 ∧ 𝜑)

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1558  ∃wex 1799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800

This theorem is referenced by: (None)