Definition df-alsi 42534

Description: Define "all some" applied to a top-level implication, which means ps is true whenever ph is true and there is at least one x where ph is true. (Contributed by David A. Wheeler, 20-Oct-2018.)

Assertion

Ref	Expression
df-alsi	|- ( A.! x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E. x ph ) )

Detailed syntax breakdown of Definition df-alsi

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2		wps	. . 3 wff ps
3		vx	. . 3 setvar x
4	1, 2, 3	walsi 42532	. 2 wff A.! x ( ph -> ps )
5	1, 2	wi 4	. . . 4 wff ( ph -> ps )
6	5, 3	wal 1481	. . 3 wff A. x ( ph -> ps )
7	1, 3	wex 1704	. . 3 wff E. x ph
8	6, 7	wa 384	. 2 wff ( A. x ( ph -> ps ) /\ E. x ph )
9	4, 8	wb 196	1 wff ( A.! x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E. x ph ) )

This definition is referenced by:  alsconv  42536  alsi1d  42537  alsi2d  42538  alsi-no-surprise  42542