New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  disamis GIF version

Theorem disamis 2314
 Description: "Disamis", one of the syllogisms of Aristotelian logic. Some φ is ψ, and all φ is χ, therefore some χ is ψ. (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
Hypotheses
Ref Expression
disamis.maj x(φ ψ)
disamis.min x(φχ)
Assertion
Ref Expression
disamis x(χ ψ)

Proof of Theorem disamis
StepHypRef Expression
1 disamis.maj . 2 x(φ ψ)
2 disamis.min . . . . 5 x(φχ)
32spi 1753 . . . 4 (φχ)
43anim1i 551 . . 3 ((φ ψ) → (χ ψ))
54eximi 1576 . 2 (x(φ ψ) → x(χ ψ))
61, 5ax-mp 8 1 x(χ ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-an 360  df-ex 1542 This theorem is referenced by:  bocardo  2316
 Copyright terms: Public domain W3C validator