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Theorem nfabdw 2930
Description: Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2932 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by Gino Giotto, 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 1917 . 2 𝑧𝜑
2 df-clab 2716 . . . 4 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
3 sb6 2088 . . . 4 ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧𝜓))
42, 3bitri 274 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ ∀𝑦(𝑦 = 𝑧𝜓))
5 nfabdw.1 . . . 4 𝑦𝜑
6 nfvd 1918 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
7 nfabdw.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
86, 7nfimd 1897 . . . 4 (𝜑 → Ⅎ𝑥(𝑦 = 𝑧𝜓))
95, 8nfald 2322 . . 3 (𝜑 → Ⅎ𝑥𝑦(𝑦 = 𝑧𝜓))
104, 9nfxfrd 1856 . 2 (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦𝜓})
111, 10nfcd 2895 1 (𝜑𝑥{𝑦𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1537  wnf 1786  [wsb 2067  wcel 2106  {cab 2715  wnfc 2887
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-10 2137  ax-11 2154  ax-12 2171
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-nfc 2889
This theorem is referenced by:  nfrabw  3318  nfrabwOLD  3319  nfsbcdw  3737  nfcsb1d  3855  nfcsbw  3859  nfifd  4488  nfunid  4845  nfopabd  5142  nfiotadw  6394  nfintd  46379  nfiund  46380
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