Theorem celarent 2113

Description: "Celarent", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore no 𝜒 is 𝜓. (In Aristotelian notation, EAE-1: MeP and SaM therefore SeP.) For example, given the "No reptiles have fur" and "All snakes are reptiles", therefore "No snakes have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

Ref	Expression
celarent.maj	⊢ ∀𝑥(𝜑 → ¬ 𝜓)
celarent.min	⊢ ∀𝑥(𝜒 → 𝜑)

Assertion

Ref	Expression
celarent	⊢ ∀𝑥(𝜒 → ¬ 𝜓)

Proof of Theorem celarent

Step	Hyp	Ref	Expression
1		celarent.maj	. 2 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
2		celarent.min	. 2 ⊢ ∀𝑥(𝜒 → 𝜑)
3	1, 2	barbara 2112	1 ⊢ ∀𝑥(𝜒 → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437

This theorem is referenced by: (None)