MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfsbcdw Structured version   Visualization version   GIF version

Theorem nfsbcdw 3808
Description: Deduction version of nfsbcw 3809. Version of nfsbcd 3811 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2376. (Revised by GG, 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 3788 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
2 nfsbcdw.2 . . 3 (𝜑𝑥𝐴)
3 nfsbcdw.1 . . . 4 𝑦𝜑
4 nfsbcdw.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
53, 4nfabdw 2926 . . 3 (𝜑𝑥{𝑦𝜓})
62, 5nfeld 2916 . 2 (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦𝜓})
71, 6nfxfrd 1853 1 (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1782  wcel 2107  {cab 2713  wnfc 2889  [wsbc 3787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-sbc 3788
This theorem is referenced by:  nfsbcw  3809  nfcsbw  3924  sbcnestgfw  4420
  Copyright terms: Public domain W3C validator