Theorem camestros 2722

Description: "Camestros", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, no 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, AEO-2: PaM and SeM therefore SoP.) For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

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

Assertion

Ref	Expression
camestros	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem camestros

Step	Hyp	Ref	Expression
1		camestros.e	. 2 ⊢ ∃𝑥𝜒
2		camestros.min	. . . . 5 ⊢ ∀𝑥(𝜒 → ¬ 𝜓)
3	2	spi 2207	. . . 4 ⊢ (𝜒 → ¬ 𝜓)
4		camestros.maj	. . . . 5 ⊢ ∀𝑥(𝜑 → 𝜓)
5	4	spi 2207	. . . 4 ⊢ (𝜑 → 𝜓)
6	3, 5	nsyl 137	. . 3 ⊢ (𝜒 → ¬ 𝜑)
7	6	ancli 530	. 2 ⊢ (𝜒 → (𝜒 ∧ ¬ 𝜑))
8	1, 7	eximii 1911	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382  ∀wal 1628  ∃wex 1851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-12 2202

This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852

This theorem is referenced by: (None)