Theorem darapti 2318

Description: "Darapti", one of the syllogisms of Aristotelian logic. All φ is ψ, all φ is χ, and some φ exist, therefore some χ is ψ. (In Aristotelian notation, AAI-3: MaP and MaS therefore SiP.) For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

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

Assertion

Ref	Expression
darapti	⊢ ∃x(χ ∧ ψ)

Proof of Theorem darapti

Step	Hyp	Ref	Expression
1		darapti.e	. 2 ⊢ ∃xφ
2		darapti.min	. . . . 5 ⊢ ∀x(φ → χ)
3	2	spi 1753	. . . 4 ⊢ (φ → χ)
4		darapti.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:   → 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)