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Definition df-bj-gab 35130
Description: Definition of generalized class abstractions: typically, 𝑥 is a bound variable in 𝐴 and 𝜑 and {𝐴𝑥𝜑} denotes "the class of 𝐴(𝑥)'s such that 𝜑(𝑥)". (Contributed by BJ, 4-Oct-2024.)
Assertion
Ref Expression
df-bj-gab {𝐴𝑥𝜑} = {𝑦 ∣ ∃𝑥(𝐴 = 𝑦𝜑)}
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Detailed syntax breakdown of Definition df-bj-gab
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
3 cA . . 3 class 𝐴
41, 2, 3bj-cgab 35129 . 2 class {𝐴𝑥𝜑}
5 vy . . . . . . 7 setvar 𝑦
65cv 1538 . . . . . 6 class 𝑦
73, 6wceq 1539 . . . . 5 wff 𝐴 = 𝑦
87, 1wa 396 . . . 4 wff (𝐴 = 𝑦𝜑)
98, 2wex 1782 . . 3 wff 𝑥(𝐴 = 𝑦𝜑)
109, 5cab 2715 . 2 class {𝑦 ∣ ∃𝑥(𝐴 = 𝑦𝜑)}
114, 10wceq 1539 1 wff {𝐴𝑥𝜑} = {𝑦 ∣ ∃𝑥(𝐴 = 𝑦𝜑)}
Colors of variables: wff setvar class
This definition is referenced by:  bj-gabss  35131  bj-gabeqis  35134  bj-elgab  35135
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