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Mirrors > Home > ILE Home > Th. List > ax-coll | GIF version |
Description: Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 4150 but uses a freeness hypothesis in place of one of the distinct variable conditions. (Contributed by Jim Kingdon, 23-Aug-2018.) |
Ref | Expression |
---|---|
ax-coll.1 | ⊢ Ⅎ𝑏𝜑 |
Ref | Expression |
---|---|
ax-coll | ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . . 4 wff 𝜑 | |
2 | vy | . . . 4 setvar 𝑦 | |
3 | 1, 2 | wex 1480 | . . 3 wff ∃𝑦𝜑 |
4 | vx | . . 3 setvar 𝑥 | |
5 | va | . . . 4 setvar 𝑎 | |
6 | 5 | cv 1342 | . . 3 class 𝑎 |
7 | 3, 4, 6 | wral 2443 | . 2 wff ∀𝑥 ∈ 𝑎 ∃𝑦𝜑 |
8 | vb | . . . . . 6 setvar 𝑏 | |
9 | 8 | cv 1342 | . . . . 5 class 𝑏 |
10 | 1, 2, 9 | wrex 2444 | . . . 4 wff ∃𝑦 ∈ 𝑏 𝜑 |
11 | 10, 4, 6 | wral 2443 | . . 3 wff ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 |
12 | 11, 8 | wex 1480 | . 2 wff ∃𝑏∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 |
13 | 7, 12 | wi 4 | 1 wff (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑) |
Colors of variables: wff set class |
This axiom is referenced by: repizf 4097 bnd 4150 |
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