Theorem cesare 2557

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 2552. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)

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	. . . 4 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
2	1	spi 2042	. . 3 ⊢ (𝜑 → ¬ 𝜓)
3		cesare.min	. . . 4 ⊢ ∀𝑥(𝜒 → 𝜓)
4	3	spi 2042	. . 3 ⊢ (𝜒 → 𝜓)
5	2, 4	nsyl3 132	. 2 ⊢ (𝜒 → ¬ 𝜑)
6	5	ax-gen 1713	1 ⊢ ∀𝑥(𝜒 → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by: (None)