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	⊢ 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 1917	. 2 ⊢ Ⅎ𝑧𝜑
2		df-clab 2716	. . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
3		sb6 2088	. . . 4 ⊢ ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓))
4	2, 3	bitri 274	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓))
5		nfabdw.1	. . . 4 ⊢ Ⅎ𝑦𝜑
6		nfvd 1918	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
7		nfabdw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
8	6, 7	nfimd 1897	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = 𝑧 → 𝜓))
9	5, 8	nfald 2322	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 = 𝑧 → 𝜓))
10	4, 9	nfxfrd 1856	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓})
11	1, 10	nfcd 2895	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Syntax hints:   → wi 4  ∀wal 1537  Ⅎwnf 1786  [wsb 2067   ∈ wcel 2106  {cab 2715  Ⅎwnfc 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