Theorem cesare 2307

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 2302. (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 1753	. . 3 ⊢ (φ → ¬ ψ)
3		cesare.min	. . . 4 ⊢ ∀x(χ → ψ)
4	3	spi 1753	. . 3 ⊢ (χ → ψ)
5	2, 4	nsyl3 111	. 2 ⊢ (χ → ¬ φ)
6	5	ax-gen 1546	1 ⊢ ∀x(χ → ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by: (None)