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Theorem bamalip 2324
 Description: "Bamalip", one of the syllogisms of Aristotelian logic. All φ is ψ, all ψ is χ, and φ exist, therefore some χ is φ. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2305. (Contributed by David A. Wheeler, 28-Aug-2016.)
Hypotheses
Ref Expression
bamalip.maj x(φψ)
bamalip.min x(ψχ)
bamalip.e xφ
Assertion
Ref Expression
bamalip x(χ φ)

Proof of Theorem bamalip
StepHypRef Expression
1 bamalip.e . 2 xφ
2 bamalip.maj . . . . . 6 x(φψ)
32spi 1753 . . . . 5 (φψ)
4 bamalip.min . . . . . 6 x(ψχ)
54spi 1753 . . . . 5 (ψχ)
63, 5syl 15 . . . 4 (φχ)
76ancri 535 . . 3 (φ → (χ φ))
87eximi 1576 . 2 (xφx(χ φ))
91, 8ax-mp 8 1 x(χ φ)
 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: (None)
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