Definition df-als 50717

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 50715	. 2 wff ∀∃𝑥(𝜑 → 𝜓)
5	1, 2	wi 4	. . . 4 wff (𝜑 → 𝜓)
6	5, 3	wal 1568	. . 3 wff ∀𝑥(𝜑 → 𝜓)
7	1, 3	wex 1812	. . 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  50719  alsd  50720  als1d  50722  als2d  50723  alsex  50727  alsbii  50729  alsbid  50731  nfals  50732  cbvals  50734  als-no-surprise  50735  alsanmo  50739  alsralrex  50741  alseuals  50753