Theorem cesaro 2127

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

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	. . . . 5 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
3	2	spi 1529	. . . 4 ⊢ (𝜑 → ¬ 𝜓)
4		cesaro.min	. . . . 5 ⊢ ∀𝑥(𝜒 → 𝜓)
5	4	spi 1529	. . . 4 ⊢ (𝜒 → 𝜓)
6	3, 5	nsyl3 621	. . 3 ⊢ (𝜒 → ¬ 𝜑)
7	6	ancli 321	. 2 ⊢ (𝜒 → (𝜒 ∧ ¬ 𝜑))
8	1, 7	eximii 1595	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1346  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)