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Theorem bamalip 2007
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 1988. (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 . . . . 5 x(φψ)
32spi 1417 . . . 4 (φψ)
4 bamalip.min . . . . 5 x(ψχ)
54spi 1417 . . . 4 (ψχ)
63, 5syl 14 . . 3 (φχ)
76ancri 307 . 2 (φ → (χ φ))
81, 7eximii 1482 1 x(χ φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1231  wex 1368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1321  ax-gen 1323  ax-ie1 1369  ax-ie2 1370  ax-4 1387  ax-ial 1415
This theorem depends on definitions:  df-bi 110
This theorem is referenced by: (None)
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