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Theorem nfabd 2917
Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2365. Use the weaker nfabdw 2915 when possible. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2108 and ax-ext 2696. (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 1909 . 2 𝑧𝜑
2 df-clab 2703 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
3 nfabd.1 . . . 4 𝑦𝜑
4 nfabd.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
53, 4nfsbd 2515 . . 3 (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
62, 5nfxfrd 1848 . 2 (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦𝜓})
71, 6nfcd 2883 1 (𝜑𝑥{𝑦𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1777  [wsb 2059  wcel 2098  {cab 2702  wnfc 2875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2166  ax-13 2365
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-nfc 2877
This theorem is referenced by:  nfabd2  2918  nfsbcd  3799  nfcsbd  3917  nfiotad  6510  nfiundg  48358
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