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Axiom ax-coll 3919
Description: Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 3972 but uses a freeness hypothesis in place of one of the distinct variable constraints. (Contributed by Jim Kingdon, 23-Aug-2018.)
Hypothesis
Ref Expression
ax-coll.1 𝑏𝜑
Assertion
Ref Expression
ax-coll (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)
Distinct variable group:   𝑥,𝑦,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏)

Detailed syntax breakdown of Axiom ax-coll
StepHypRef Expression
1 wph . . . 4 wff 𝜑
2 vy . . . 4 setvar 𝑦
31, 2wex 1422 . . 3 wff 𝑦𝜑
4 vx . . 3 setvar 𝑥
5 va . . . 4 setvar 𝑎
65cv 1284 . . 3 class 𝑎
73, 4, 6wral 2353 . 2 wff 𝑥𝑎𝑦𝜑
8 vb . . . . . 6 setvar 𝑏
98cv 1284 . . . . 5 class 𝑏
101, 2, 9wrex 2354 . . . 4 wff 𝑦𝑏 𝜑
1110, 4, 6wral 2353 . . 3 wff 𝑥𝑎𝑦𝑏 𝜑
1211, 8wex 1422 . 2 wff 𝑏𝑥𝑎𝑦𝑏 𝜑
137, 12wi 4 1 wff (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)
Colors of variables: wff set class
This axiom is referenced by:  repizf  3920  bnd  3972
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