Theorem bamalip 2586

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 2567. (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

Step	Hyp	Ref	Expression
1		bamalip.e	. 2 |- E. x ph
2		bamalip.maj	. . . . 5 |- A. x ( ph -> ps )
3	2	spi 2054	. . . 4 |- ( ph -> ps )
4		bamalip.min	. . . . 5 |- A. x ( ps -> ch )
5	4	spi 2054	. . . 4 |- ( ps -> ch )
6	3, 5	syl 17	. . 3 |- ( ph -> ch )
7	6	ancri 575	. 2 |- ( ph -> ( ch /\ ph ) )
8	1, 7	eximii 1764	1 |- E. x ( ch /\ ph )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)