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Theorem nfabdw 2925
Description: Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2926 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-13 2375. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Sep-2024.)
Hypotheses
Ref Expression
nfabdw.1 𝑦𝜑
nfabdw.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfabdw (𝜑𝑥{𝑦𝜓})
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem nfabdw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1912 . 2 𝑧𝜑
2 df-clab 2713 . . . 4 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
3 sb6 2083 . . . 4 ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧𝜓))
42, 3bitri 275 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ ∀𝑦(𝑦 = 𝑧𝜓))
5 nfabdw.1 . . . 4 𝑦𝜑
6 nfvd 1913 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
7 nfabdw.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
86, 7nfimd 1892 . . . 4 (𝜑 → Ⅎ𝑥(𝑦 = 𝑧𝜓))
95, 8nfald 2327 . . 3 (𝜑 → Ⅎ𝑥𝑦(𝑦 = 𝑧𝜓))
104, 9nfxfrd 1851 . 2 (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦𝜓})
111, 10nfcd 2896 1 (𝜑𝑥{𝑦𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535  wnf 1780  [wsb 2062  wcel 2106  {cab 2712  wnfc 2888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-nfc 2890
This theorem is referenced by:  nfrabw  3473  nfrabwOLD  3474  nfsbcdw  3812  nfcsb1d  3931  nfcsbw  3935  nfifd  4560  nfunid  4918  nfopabd  5216  nfiotadw  6519  nfintd  48904  nfiund  48905
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