Definition df-alsi 42534

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

Assertion

Ref	Expression
df-alsi	⊢ (∀!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))

Detailed syntax breakdown of Definition df-alsi

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		wps	. . 3 wff 𝜓
3		vx	. . 3 setvar 𝑥
4	1, 2, 3	walsi 42532	. 2 wff ∀!𝑥(𝜑 → 𝜓)
5	1, 2	wi 4	. . . 4 wff (𝜑 → 𝜓)
6	5, 3	wal 1481	. . 3 wff ∀𝑥(𝜑 → 𝜓)
7	1, 3	wex 1704	. . 3 wff ∃𝑥𝜑
8	6, 7	wa 384	. 2 wff (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑)
9	4, 8	wb 196	1 wff (∀!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))

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