Theorem calemos 2572

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.) (Revised by David A. Wheeler, 2-Sep-2016.)

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.min	. . . . . 6 ⊢ ∀𝑥(𝜓 → ¬ 𝜒)
3	2	spi 2042	. . . . 5 ⊢ (𝜓 → ¬ 𝜒)
4	3	con2i 133	. . . 4 ⊢ (𝜒 → ¬ 𝜓)
5		calemos.maj	. . . . 5 ⊢ ∀𝑥(𝜑 → 𝜓)
6	5	spi 2042	. . . 4 ⊢ (𝜑 → 𝜓)
7	4, 6	nsyl 134	. . 3 ⊢ (𝜒 → ¬ 𝜑)
8	7	ancli 572	. 2 ⊢ (𝜒 → (𝜒 ∧ ¬ 𝜑))
9	1, 8	eximii 1754	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by: (None)