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Theorem nfsbcdw 3065
Description: Version of nfsbcd 2956 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
nfsbcdw.1 𝑦𝜑
nfsbcdw.2 (𝜑𝑥𝐴)
nfsbcdw.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfsbcdw (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfsbcdw
StepHypRef Expression
1 df-sbc 2938 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
2 nfsbcdw.2 . . 3 (𝜑𝑥𝐴)
3 nfsbcdw.1 . . . 4 𝑦𝜑
4 nfsbcdw.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
53, 4nfabdw 2318 . . 3 (𝜑𝑥{𝑦𝜓})
62, 5nfeld 2315 . 2 (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦𝜓})
71, 6nfxfrd 1455 1 (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1440  wcel 2128  {cab 2143  wnfc 2286  [wsbc 2937
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-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-sbc 2938
This theorem is referenced by:  nfcsbw  3067
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