Theorem nfabdw 2921

Description: Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2922 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Sep-2024.)

Hypotheses

Ref	Expression
nfabdw.1	⊢ Ⅎ𝑦𝜑
nfabdw.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfabdw	⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

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.

Step	Hyp	Ref	Expression
1		nfv 1914	. 2 ⊢ Ⅎ𝑧𝜑
2		df-clab 2715	. . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
3		sb6 2086	. . . 4 ⊢ ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓))
4	2, 3	bitri 275	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓))
5		nfabdw.1	. . . 4 ⊢ Ⅎ𝑦𝜑
6		nfvd 1915	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
7		nfabdw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
8	6, 7	nfimd 1894	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = 𝑧 → 𝜓))
9	5, 8	nfald 2329	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 = 𝑧 → 𝜓))
10	4, 9	nfxfrd 1854	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓})
11	1, 10	nfcd 2892	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Syntax hints:   → wi 4  ∀wal 1538  Ⅎwnf 1783  [wsb 2065   ∈ wcel 2109  {cab 2714  Ⅎwnfc 2884

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 2008  ax-10 2142  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-nfc 2886

This theorem is referenced by:  nfrabw  3459  nfrabwOLD  3460  nfsbcdw  3791  nfcsb1d  3901  nfcsbw  3905  nfifd  4535  nfunid  4894  nfopabd  5192  nfiotadw  6492  nfintd  49504  nfiund  49505