Theorem fresison 2321

Description: "Fresison", one of the syllogisms of Aristotelian logic. No φ is ψ (PeM), and some ψ is χ (MiS), therefore some χ is not φ (SoP). (In Aristotelian notation, EIO-4: PeM and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

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

Assertion

Ref	Expression
fresison	⊢ ∃x(χ ∧ ¬ φ)

Proof of Theorem fresison

Step	Hyp	Ref	Expression
1		fresison.min	. 2 ⊢ ∃x(ψ ∧ χ)
2		simpr 447	. . . 4 ⊢ ((ψ ∧ χ) → χ)
3		fresison.maj	. . . . . . 7 ⊢ ∀x(φ → ¬ ψ)
4	3	spi 1753	. . . . . 6 ⊢ (φ → ¬ ψ)
5	4	con2i 112	. . . . 5 ⊢ (ψ → ¬ φ)
6	5	adantr 451	. . . 4 ⊢ ((ψ ∧ χ) → ¬ φ)
7	2, 6	jca 518	. . 3 ⊢ ((ψ ∧ χ) → (χ ∧ ¬ φ))
8	7	eximi 1576	. 2 ⊢ (∃x(ψ ∧ χ) → ∃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)