Theorem cesaro 2680

Description: "Cesaro", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. In Aristotelian notation, EAO-2: PeM and SaM therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)

Hypotheses

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

Assertion

Ref	Expression
cesaro	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem cesaro

Step	Hyp	Ref	Expression
1		cesaro.e	. 2 ⊢ ∃𝑥𝜒
2		cesaro.maj	. . 3 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
3		cesaro.min	. . 3 ⊢ ∀𝑥(𝜒 → 𝜓)
4	2, 3	cesare 2674	. 2 ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
5	1, 4	barbarilem 2670	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1539  ∃wex 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786

This theorem is referenced by: (None)