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Axiom ax-bdsetind 7186
Description: Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
Hypothesis
Ref Expression
ax-bdsetind.bd BOUNDED φ
Assertion
Ref Expression
ax-bdsetind (𝑎(y 𝑎 [y / 𝑎]φφ) → 𝑎φ)
Distinct variable groups:   y,𝑎   φ,y
Allowed substitution hint:   φ(𝑎)

Detailed syntax breakdown of Axiom ax-bdsetind
StepHypRef Expression
1 wph . . . . . 6 wff φ
2 va . . . . . 6 setvar 𝑎
3 vy . . . . . 6 setvar y
41, 2, 3wsb 1627 . . . . 5 wff [y / 𝑎]φ
52cv 1227 . . . . 5 class 𝑎
64, 3, 5wral 2284 . . . 4 wff y 𝑎 [y / 𝑎]φ
76, 1wi 4 . . 3 wff (y 𝑎 [y / 𝑎]φφ)
87, 2wal 1226 . 2 wff 𝑎(y 𝑎 [y / 𝑎]φφ)
91, 2wal 1226 . 2 wff 𝑎φ
108, 9wi 4 1 wff (𝑎(y 𝑎 [y / 𝑎]φφ) → 𝑎φ)
Colors of variables: wff set class
This axiom is referenced by:  bdsetindis  7187
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