Theorem calemes 2161

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.) (Revised by David A. Wheeler, 2-Sep-2016.)

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.min	. . . . 5 ⊢ ∀𝑥(𝜓 → ¬ 𝜒)
2	1	spi 1550	. . . 4 ⊢ (𝜓 → ¬ 𝜒)
3	2	con2i 628	. . 3 ⊢ (𝜒 → ¬ 𝜓)
4		calemes.maj	. . . 4 ⊢ ∀𝑥(𝜑 → 𝜓)
5	4	spi 1550	. . 3 ⊢ (𝜑 → 𝜓)
6	3, 5	nsyl 629	. 2 ⊢ (𝜒 → ¬ 𝜑)
7	6	ax-gen 1463	1 ⊢ ∀𝑥(𝜒 → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 615  ax-in2 616  ax-gen 1463  ax-4 1524

This theorem is referenced by: (None)