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Theorem nfabd 2928
Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker nfabdw 2927 when possible. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2118 and ax-ext 2708. (Revised by Wolf Lammen, 23-May-2023.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfabd.1 𝑦𝜑
nfabd.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfabd (𝜑𝑥{𝑦𝜓})

Proof of Theorem nfabd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1914 . 2 𝑧𝜑
2 df-clab 2715 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
3 nfabd.1 . . . 4 𝑦𝜑
4 nfabd.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
53, 4nfsbd 2527 . . 3 (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
62, 5nfxfrd 1854 . 2 (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦𝜓})
71, 6nfcd 2898 1 (𝜑𝑥{𝑦𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1783  [wsb 2064  wcel 2108  {cab 2714  wnfc 2890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-nfc 2892
This theorem is referenced by:  nfabd2  2929  nfsbcd  3812  nfcsbd  3924  nfiotad  6519  nfiundg  49194
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