Theorem cesare 2672

Description: "Cesare", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜓, therefore no 𝜒 is 𝜑. In Aristotelian notation, EAE-2: PeM and SaM therefore SeP. Related to celarent 2664. (Contributed by David A. Wheeler, 27-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)

Hypotheses

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

Assertion

Ref	Expression
cesare	⊢ ∀𝑥(𝜒 → ¬ 𝜑)

Proof of Theorem cesare

Step	Hyp	Ref	Expression
1		cesare.maj	. . 3 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
2		con2 135	. . . 4 ⊢ ((𝜑 → ¬ 𝜓) → (𝜓 → ¬ 𝜑))
3	2	alimi 1811	. . 3 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ∀𝑥(𝜓 → ¬ 𝜑))
4	1, 3	ax-mp 5	. 2 ⊢ ∀𝑥(𝜓 → ¬ 𝜑)
5		cesare.min	. 2 ⊢ ∀𝑥(𝜒 → 𝜓)
6	4, 5	celarent 2664	1 ⊢ ∀𝑥(𝜒 → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538

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

This theorem is referenced by:  cesaro  2678