Theorem celarent 2302

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 2301	1 ⊢ ∀x(χ → ¬ ψ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by: (None)