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Theorem setindf 15102
Description: Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.)
Hypothesis
Ref Expression
setindf.nf 𝑦𝜑
Assertion
Ref Expression
setindf (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem setindf
StepHypRef Expression
1 setindft 15101 . 2 (∀𝑥𝑦𝜑 → (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥𝜑))
2 setindf.nf . 2 𝑦𝜑
31, 2mpg 1462 1 (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362  wnf 1471  [wsb 1773  wral 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-ral 2473
This theorem is referenced by: (None)
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