Theorem celaront 2306

Description: "Celaront", one of the syllogisms of Aristotelian logic. No φ is ψ, all χ is φ, and some χ exist, therefore some χ is not ψ. (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

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

Assertion

Ref	Expression
celaront	⊢ ∃x(χ ∧ ¬ ψ)

Proof of Theorem celaront

Step	Hyp	Ref	Expression
1		celaront.maj	. 2 ⊢ ∀x(φ → ¬ ψ)
2		celaront.min	. 2 ⊢ ∀x(χ → φ)
3		celaront.e	. 2 ⊢ ∃xχ
4	1, 2, 3	barbari 2305	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)