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Theorem nfsbcdw 3737
Description: Deduction version of nfsbcw 3738. Version of nfsbcd 3740 with a disjoint variable condition, which does not require ax-13 2372. (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 3717 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
2 nfsbcdw.2 . . 3 (𝜑𝑥𝐴)
3 nfsbcdw.1 . . . 4 𝑦𝜑
4 nfsbcdw.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
53, 4nfabdw 2930 . . 3 (𝜑𝑥{𝑦𝜓})
62, 5nfeld 2918 . 2 (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦𝜓})
71, 6nfxfrd 1856 1 (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1786  wcel 2106  {cab 2715  wnfc 2887  [wsbc 3716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-sbc 3717
This theorem is referenced by:  nfsbcw  3738  nfcsbw  3859  sbcnestgfw  4352
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