Theorem celaront 2698

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 2696. 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 2696. (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 2696	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1568  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is referenced by: (None)