Theorem celaront 2568

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

Step	Hyp	Ref	Expression
1		celaront.maj	. 2 |- A. x ( ph -> -. ps )
2		celaront.min	. 2 |- A. x ( ch -> ph )
3		celaront.e	. 2 |- E. x ch
4	1, 2, 3	barbari 2567	1 |- E. x ( ch /\ -. ps )

Syntax hints:   -. wn 3   -> 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)