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Definition df-alsc 42344
Description: Define "all some" applied to a class, which means 𝜑 is true for all 𝑥 in 𝐴 and there is at least one 𝑥 in 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.)
Assertion
Ref Expression
df-alsc (∀!𝑥𝐴𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∃𝑥 𝑥𝐴))

Detailed syntax breakdown of Definition df-alsc
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
3 cA . . 3 class 𝐴
41, 2, 3walsc 42342 . 2 wff ∀!𝑥𝐴𝜑
51, 2, 3wral 2896 . . 3 wff 𝑥𝐴 𝜑
62cv 1474 . . . . 5 class 𝑥
76, 3wcel 1977 . . . 4 wff 𝑥𝐴
87, 2wex 1695 . . 3 wff 𝑥 𝑥𝐴
95, 8wa 383 . 2 wff (∀𝑥𝐴 𝜑 ∧ ∃𝑥 𝑥𝐴)
104, 9wb 195 1 wff (∀!𝑥𝐴𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∃𝑥 𝑥𝐴))
Colors of variables: wff setvar class
This definition is referenced by:  alsconv  42345  alsc1d  42348  alsc2d  42349
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