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Theorem bamalip 2374
Description: "Bamalip", one of the syllogisms of Aristotelian logic. All  ph is  ps, all  ps is  ch, and  ph exist, therefore some  ch is  ph. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2355. (Contributed by David A. Wheeler, 28-Aug-2016.)
Hypotheses
Ref Expression
bamalip.maj  |-  A. x
( ph  ->  ps )
bamalip.min  |-  A. x
( ps  ->  ch )
bamalip.e  |-  E. x ph
Assertion
Ref Expression
bamalip  |-  E. x
( ch  /\  ph )

Proof of Theorem bamalip
StepHypRef Expression
1 bamalip.e . 2  |-  E. x ph
2 bamalip.maj . . . . 5  |-  A. x
( ph  ->  ps )
32spi 1765 . . . 4  |-  ( ph  ->  ps )
4 bamalip.min . . . . 5  |-  A. x
( ps  ->  ch )
54spi 1765 . . . 4  |-  ( ps 
->  ch )
63, 5syl 16 . . 3  |-  ( ph  ->  ch )
76ancri 536 . 2  |-  ( ph  ->  ( ch  /\  ph ) )
81, 7eximii 1584 1  |-  E. x
( ch  /\  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546   E.wex 1547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-11 1757
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548
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