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Axiom ax-bdsep 7111
Description: Axiom scheme of bounded (or restricted, or Δ0) separation. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. For the full axiom of separation, see ax-sep 3849. (Contributed by BJ, 5-Oct-2019.)
Hypothesis
Ref Expression
ax-bdsep.1 BOUNDED φ
Assertion
Ref Expression
ax-bdsep 𝑎𝑏x(x 𝑏 ↔ (x 𝑎 φ))
Distinct variable groups:   𝑎,𝑏,x   φ,𝑎,𝑏
Allowed substitution hint:   φ(x)

Detailed syntax breakdown of Axiom ax-bdsep
StepHypRef Expression
1 vx . . . . . 6 setvar x
2 vb . . . . . 6 setvar 𝑏
31, 2wel 1375 . . . . 5 wff x 𝑏
4 va . . . . . . 7 setvar 𝑎
51, 4wel 1375 . . . . . 6 wff x 𝑎
6 wph . . . . . 6 wff φ
75, 6wa 97 . . . . 5 wff (x 𝑎 φ)
83, 7wb 98 . . . 4 wff (x 𝑏 ↔ (x 𝑎 φ))
98, 1wal 1226 . . 3 wff x(x 𝑏 ↔ (x 𝑎 φ))
109, 2wex 1362 . 2 wff 𝑏x(x 𝑏 ↔ (x 𝑎 φ))
1110, 4wal 1226 1 wff 𝑎𝑏x(x 𝑏 ↔ (x 𝑎 φ))
Colors of variables: wff set class
This axiom is referenced by:  bdsep2  7112
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