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Theorem nfsbcdw 3083
Description: Version of nfsbcd 2974 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 2956 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
2 nfsbcdw.2 . . 3 (𝜑𝑥𝐴)
3 nfsbcdw.1 . . . 4 𝑦𝜑
4 nfsbcdw.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
53, 4nfabdw 2331 . . 3 (𝜑𝑥{𝑦𝜓})
62, 5nfeld 2328 . 2 (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦𝜓})
71, 6nfxfrd 1468 1 (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1453  wcel 2141  {cab 2156  wnfc 2299  [wsbc 2955
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-sbc 2956
This theorem is referenced by:  nfcsbw  3085
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