Definition df-als 51067

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-als	⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))

Detailed syntax breakdown of Definition df-als

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		wps	. . 3 wff 𝜓
3		vx	. . 3 setvar 𝑥
4	1, 2, 3	wals 51065	. 2 wff ∀∃𝑥(𝜑 → 𝜓)
5	1, 2	wi 4	. . . 4 wff (𝜑 → 𝜓)
6	5, 3	wal 1567	. . 3 wff ∀𝑥(𝜑 → 𝜓)
7	1, 3	wex 1811	. . 3 wff ∃𝑥𝜑
8	6, 7	wa 401	. 2 wff (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑)
9	4, 8	wb 209	1 wff (∀∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))

This definition is used by:  dfrals2  51069  alsd  51070  als1d  51072  als2d  51073  alsex  51077  alsbii  51079  alsbid  51081  nfals  51082  cbvals  51084  als-no-surprise  51085  alsanmo  51089  alsralrex  51091  alseuals  51103