Theorem celaront 2671

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 2669. 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 2669. (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 2669	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by: (None)