Theorem disamis 2706

Description: "Disamis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. In Aristotelian notation, IAI-3: MiP and MaS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)

Hypotheses

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

Assertion

Ref	Expression
disamis	⊢ ∃𝑥(𝜒 ∧ 𝜓)

Proof of Theorem disamis

Step	Hyp	Ref	Expression
1		disamis.min	. . 3 ⊢ ∀𝑥(𝜑 → 𝜒)
2		disamis.maj	. . 3 ⊢ ∃𝑥(𝜑 ∧ 𝜓)
3	1, 2	datisi 2705	. 2 ⊢ ∃𝑥(𝜓 ∧ 𝜒)
4		exancom 1894	. 2 ⊢ (∃𝑥(𝜓 ∧ 𝜒) ↔ ∃𝑥(𝜒 ∧ 𝜓))
5	3, 4	mpbi 233	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  bocardo  2708