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Theorem setindft 10918
Description: Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
Assertion
Ref Expression
setindft (∀𝑥𝑦𝜑 → (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem setindft
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfa1 1475 . . 3 𝑥𝑥𝑦𝜑
2 nfv 1462 . . . . . 6 𝑧𝑥𝑦𝜑
3 nfnf1 1477 . . . . . . 7 𝑦𝑦𝜑
43nfal 1509 . . . . . 6 𝑦𝑥𝑦𝜑
5 nfsbt 1892 . . . . . 6 (∀𝑥𝑦𝜑 → Ⅎ𝑦[𝑧 / 𝑥]𝜑)
6 nfv 1462 . . . . . . 7 𝑧[𝑦 / 𝑥]𝜑
76a1i 9 . . . . . 6 (∀𝑥𝑦𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
8 sbequ 1762 . . . . . . 7 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
98a1i 9 . . . . . 6 (∀𝑥𝑦𝜑 → (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)))
102, 4, 5, 7, 9cbvrald 10749 . . . . 5 (∀𝑥𝑦𝜑 → (∀𝑧𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝑥 [𝑦 / 𝑥]𝜑))
1110biimpd 142 . . . 4 (∀𝑥𝑦𝜑 → (∀𝑧𝑥 [𝑧 / 𝑥]𝜑 → ∀𝑦𝑥 [𝑦 / 𝑥]𝜑))
1211imim1d 74 . . 3 (∀𝑥𝑦𝜑 → ((∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → (∀𝑧𝑥 [𝑧 / 𝑥]𝜑𝜑)))
131, 12alimd 1455 . 2 (∀𝑥𝑦𝜑 → (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥(∀𝑧𝑥 [𝑧 / 𝑥]𝜑𝜑)))
14 ax-setind 4288 . 2 (∀𝑥(∀𝑧𝑥 [𝑧 / 𝑥]𝜑𝜑) → ∀𝑥𝜑)
1513, 14syl6 33 1 (∀𝑥𝑦𝜑 → (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283  wnf 1390  [wsb 1686  wral 2349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078  df-ral 2354
This theorem is referenced by:  setindf  10919
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