Theorem calemes 2319

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 1753	. . . 4 ⊢ (ψ → ¬ χ)
3	2	con2i 112	. . 3 ⊢ (χ → ¬ ψ)
4		calemes.maj	. . . 4 ⊢ ∀x(φ → ψ)
5	4	spi 1753	. . 3 ⊢ (φ → ψ)
6	3, 5	nsyl 113	. 2 ⊢ (χ → ¬ φ)
7	6	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)