Theorem celaront 2700

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 2698. 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 2698. (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 2698	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)