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Theorem setindft 10090
 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 1434 . . 3 𝑥𝑥𝑦𝜑
2 nfv 1421 . . . . . 6 𝑧𝑥𝑦𝜑
3 nfnf1 1436 . . . . . . 7 𝑦𝑦𝜑
43nfal 1468 . . . . . 6 𝑦𝑥𝑦𝜑
5 nfsbt 1850 . . . . . 6 (∀𝑥𝑦𝜑 → Ⅎ𝑦[𝑧 / 𝑥]𝜑)
6 nfv 1421 . . . . . . 7 𝑧[𝑦 / 𝑥]𝜑
76a1i 9 . . . . . 6 (∀𝑥𝑦𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
8 sbequ 1721 . . . . . . 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 9927 . . . . 5 (∀𝑥𝑦𝜑 → (∀𝑧𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝑥 [𝑦 / 𝑥]𝜑))
1110biimpd 132 . . . 4 (∀𝑥𝑦𝜑 → (∀𝑧𝑥 [𝑧 / 𝑥]𝜑 → ∀𝑦𝑥 [𝑦 / 𝑥]𝜑))
1211imim1d 69 . . 3 (∀𝑥𝑦𝜑 → ((∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → (∀𝑧𝑥 [𝑧 / 𝑥]𝜑𝜑)))
131, 12alimd 1414 . 2 (∀𝑥𝑦𝜑 → (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥(∀𝑧𝑥 [𝑧 / 𝑥]𝜑𝜑)))
14 ax-setind 4262 . 2 (∀𝑥(∀𝑧𝑥 [𝑧 / 𝑥]𝜑𝜑) → ∀𝑥𝜑)
1513, 14syl6 29 1 (∀𝑥𝑦𝜑 → (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349  [wsb 1645  ∀wral 2306 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-ral 2311 This theorem is referenced by:  setindf  10091
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