Theorem cesaro 2311

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	. . . . . 6 ⊢ ∀x(φ → ¬ ψ)
3	2	spi 1753	. . . . 5 ⊢ (φ → ¬ ψ)
4		cesaro.min	. . . . . 6 ⊢ ∀x(χ → ψ)
5	4	spi 1753	. . . . 5 ⊢ (χ → ψ)
6	3, 5	nsyl3 111	. . . 4 ⊢ (χ → ¬ φ)
7	6	ancli 534	. . 3 ⊢ (χ → (χ ∧ ¬ φ))
8	7	eximi 1576	. 2 ⊢ (∃xχ → ∃x(χ ∧ ¬ φ))
9	1, 8	ax-mp 5	1 ⊢ ∃x(χ ∧ ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)