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Theorem bamalip 2407
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 2388. (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 1771 . . . 4  |-  ( ph  ->  ps )
4 bamalip.min . . . . 5  |-  A. x
( ps  ->  ch )
54spi 1771 . . . 4  |-  ( ps 
->  ch )
63, 5syl 16 . . 3  |-  ( ph  ->  ch )
76ancri 537 . 2  |-  ( ph  ->  ( ch  /\  ph ) )
81, 7eximii 1588 1  |-  E. x
( ch  /\  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550   E.wex 1551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-11 1763
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552
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