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Theorem celaront 2045
Description: "Celaront", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ch is  ph, and some  ch exist, therefore some  ch is not  ps. (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
celaront.maj  |-  A. x
( ph  ->  -.  ps )
celaront.min  |-  A. x
( ch  ->  ph )
celaront.e  |-  E. x ch
Assertion
Ref Expression
celaront  |-  E. x
( ch  /\  -.  ps )

Proof of Theorem celaront
StepHypRef Expression
1 celaront.maj . 2  |-  A. x
( ph  ->  -.  ps )
2 celaront.min . 2  |-  A. x
( ch  ->  ph )
3 celaront.e . 2  |-  E. x ch
41, 2, 3barbari 2044 1  |-  E. x
( ch  /\  -.  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102   A.wal 1283   E.wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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