Theorem festino 2309

Description: "Festino", one of the syllogisms of Aristotelian logic. No φ is ψ, and some χ is ψ, therefore some χ is not φ. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)

Hypotheses

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

Assertion

Ref	Expression
festino	⊢ ∃x(χ ∧ ¬ φ)

Proof of Theorem festino

Step	Hyp	Ref	Expression
1		festino.min	. 2 ⊢ ∃x(χ ∧ ψ)
2		festino.maj	. . . . . 6 ⊢ ∀x(φ → ¬ ψ)
3	2	spi 1753	. . . . 5 ⊢ (φ → ¬ ψ)
4	3	con2i 112	. . . 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)