Theorem celaront 2697

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 2695. 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 2695. (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 2695	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1558  ∃wex 1799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800

This theorem is referenced by: (None)