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Theorem setindft 13551
Description: Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness 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 1521 . . 3 𝑥𝑥𝑦𝜑
2 nfv 1508 . . . . . 6 𝑧𝑥𝑦𝜑
3 nfnf1 1524 . . . . . . 7 𝑦𝑦𝜑
43nfal 1556 . . . . . 6 𝑦𝑥𝑦𝜑
5 nfsbt 1956 . . . . . 6 (∀𝑥𝑦𝜑 → Ⅎ𝑦[𝑧 / 𝑥]𝜑)
6 nfv 1508 . . . . . . 7 𝑧[𝑦 / 𝑥]𝜑
76a1i 9 . . . . . 6 (∀𝑥𝑦𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
8 sbequ 1820 . . . . . . 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 13373 . . . . 5 (∀𝑥𝑦𝜑 → (∀𝑧𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝑥 [𝑦 / 𝑥]𝜑))
1110biimpd 143 . . . 4 (∀𝑥𝑦𝜑 → (∀𝑧𝑥 [𝑧 / 𝑥]𝜑 → ∀𝑦𝑥 [𝑦 / 𝑥]𝜑))
1211imim1d 75 . . 3 (∀𝑥𝑦𝜑 → ((∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → (∀𝑧𝑥 [𝑧 / 𝑥]𝜑𝜑)))
131, 12alimd 1501 . 2 (∀𝑥𝑦𝜑 → (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥(∀𝑧𝑥 [𝑧 / 𝑥]𝜑𝜑)))
14 ax-setind 4496 . 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 104  wal 1333  wnf 1440  [wsb 1742  wral 2435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-setind 4496
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-ral 2440
This theorem is referenced by:  setindf  13552
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