Theorem calemes 2687

Description: "Calemes", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜓 is 𝜒, therefore no 𝜒 is 𝜑. In Aristotelian notation, AEE-4: PaM and MeS therefore SeP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)

Hypotheses

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

Assertion

Ref	Expression
calemes	⊢ ∀𝑥(𝜒 → ¬ 𝜑)

Proof of Theorem calemes

Step	Hyp	Ref	Expression
1		calemes.maj	. 2 ⊢ ∀𝑥(𝜑 → 𝜓)
2		calemes.min	. . 3 ⊢ ∀𝑥(𝜓 → ¬ 𝜒)
3		con2 135	. . . 4 ⊢ ((𝜓 → ¬ 𝜒) → (𝜒 → ¬ 𝜓))
4	3	alimi 1811	. . 3 ⊢ (∀𝑥(𝜓 → ¬ 𝜒) → ∀𝑥(𝜒 → ¬ 𝜓))
5	2, 4	ax-mp 5	. 2 ⊢ ∀𝑥(𝜒 → ¬ 𝜓)
6	1, 5	camestres 2673	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:  calemos  2690