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Axiom ax-bdsetind 10906
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(∀𝑦𝑎 [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 1686 . . . . 5 wff [𝑦 / 𝑎]𝜑
52cv 1284 . . . . 5 class 𝑎
64, 3, 5wral 2349 . . . 4 wff 𝑦𝑎 [𝑦 / 𝑎]𝜑
76, 1wi 4 . . 3 wff (∀𝑦𝑎 [𝑦 / 𝑎]𝜑𝜑)
87, 2wal 1283 . 2 wff 𝑎(∀𝑦𝑎 [𝑦 / 𝑎]𝜑𝜑)
91, 2wal 1283 . 2 wff 𝑎𝜑
108, 9wi 4 1 wff (∀𝑎(∀𝑦𝑎 [𝑦 / 𝑎]𝜑𝜑) → ∀𝑎𝜑)
Colors of variables: wff set class
This axiom is referenced by:  bdsetindis  10907
  Copyright terms: Public domain W3C validator