Theorem baroco 2310

Description: "Baroco", one of the syllogisms of Aristotelian logic. All φ is ψ, and some χ is not ψ, therefore some χ is not φ. (In Aristotelian notation, AOO-2: PaM and SoM therefore SoP.) For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative." (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

Ref	Expression
baroco.maj	⊢ ∀x(φ → ψ)
baroco.min	⊢ ∃x(χ ∧ ¬ ψ)

Assertion

Ref	Expression
baroco	⊢ ∃x(χ ∧ ¬ φ)

Proof of Theorem baroco

Step	Hyp	Ref	Expression
1		baroco.min	. 2 ⊢ ∃x(χ ∧ ¬ ψ)
2		baroco.maj	. . . . . 6 ⊢ ∀x(φ → ψ)
3	2	spi 1753	. . . . 5 ⊢ (φ → ψ)
4	3	con3i 127	. . . 4 ⊢ (¬ ψ → ¬ φ)
5	4	anim2i 552	. . 3 ⊢ ((χ ∧ ¬ ψ) → (χ ∧ ¬ φ))
6	5	eximi 1576	. 2 ⊢ (∃x(χ ∧ ¬ ψ) → ∃x(χ ∧ ¬ φ))
7	1, 6	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)