Theorem calemos 2691

Description: "Calemos", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓 (PaM), no 𝜓 is 𝜒 (MeS), and 𝜒 exist, therefore some 𝜒 is not 𝜑 (SoP). In Aristotelian notation, AEO-4: PaM and MeS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)

Hypotheses

Ref	Expression
calemos.maj	⊢ ∀𝑥(𝜑 → 𝜓)
calemos.min	⊢ ∀𝑥(𝜓 → ¬ 𝜒)
calemos.e	⊢ ∃𝑥𝜒

Assertion

Ref	Expression
calemos	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem calemos

Step	Hyp	Ref	Expression
1		calemos.e	. 2 ⊢ ∃𝑥𝜒
2		calemos.maj	. . 3 ⊢ ∀𝑥(𝜑 → 𝜓)
3		calemos.min	. . 3 ⊢ ∀𝑥(𝜓 → ¬ 𝜒)
4	2, 3	calemes 2688	. 2 ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
5	1, 4	barbarilem 2669	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783

This theorem is referenced by: (None)