Theorem felapton 2317

Description: "Felapton", one of the syllogisms of Aristotelian logic. No φ is ψ, all φ is χ, and some φ exist, therefore some χ is not ψ. (In Aristotelian notation, EAO-3: MeP and MaS therefore SoP.) For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

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

Assertion

Ref	Expression
felapton	⊢ ∃x(χ ∧ ¬ ψ)

Proof of Theorem felapton

Step	Hyp	Ref	Expression
1		felapton.e	. 2 ⊢ ∃xφ
2		felapton.min	. . . . 5 ⊢ ∀x(φ → χ)
3	2	spi 1753	. . . 4 ⊢ (φ → χ)
4		felapton.maj	. . . . 5 ⊢ ∀x(φ → ¬ ψ)
5	4	spi 1753	. . . 4 ⊢ (φ → ¬ ψ)
6	3, 5	jca 518	. . 3 ⊢ (φ → (χ ∧ ¬ ψ))
7	6	eximi 1576	. 2 ⊢ (∃xφ → ∃x(χ ∧ ¬ ψ))
8	1, 7	ax-mp 5	1 ⊢ ∃x(χ ∧ ¬ ψ)

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)