Theorem celaront 2674

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. Instance of barbari 2672. 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 2672. (Contributed by David A. Wheeler, 27-Aug-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 2672	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787

This theorem is referenced by: (None)