Theorem camestres 2673

Description: "Camestres", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜒 is 𝜓, therefore no 𝜒 is 𝜑. In Aristotelian notation, AEE-2: PaM and SeM therefore SeP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)

Hypotheses

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

Assertion

Ref	Expression
camestres	⊢ ∀𝑥(𝜒 → ¬ 𝜑)

Proof of Theorem camestres

Step	Hyp	Ref	Expression
1		camestres.maj	. . 3 ⊢ ∀𝑥(𝜑 → 𝜓)
2		con3 156	. . . 4 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
3	2	alimi 1819	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥(¬ 𝜓 → ¬ 𝜑))
4	1, 3	ax-mp 5	. 2 ⊢ ∀𝑥(¬ 𝜓 → ¬ 𝜑)
5		camestres.min	. 2 ⊢ ∀𝑥(𝜒 → ¬ 𝜓)
6	4, 5	celarent 2664	1 ⊢ ∀𝑥(𝜒 → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817

This theorem depends on definitions:  df-bi 210  df-an 400

This theorem is referenced by:  camestros  2679  calemes  2687