Theorem nfabdw 2947

Description: Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2948 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-13 2404. (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 1949	. 2 ⊢ Ⅎ𝑧𝜑
2		df-clab 2742	. . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
3		sb6 2124	. . . 4 ⊢ ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓))
4	2, 3	bitri 278	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓))
5		nfabdw.1	. . . 4 ⊢ Ⅎ𝑦𝜑
6		nfvd 1950	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
7		nfabdw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
8	6, 7	nfimd 1929	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = 𝑧 → 𝜓))
9	5, 8	nfald 2362	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 = 𝑧 → 𝜓))
10	4, 9	nfxfrd 1887	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓})
11	1, 10	nfcd 2919	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  Ⅎwnf 1816  [wsb 2101   ∈ wcel 2147  {cab 2741  Ⅎwnfc 2911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-11 2197  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-nfc 2913

This theorem is used by:  nfsbcdw  3763  nfcsb1d  3873  nfcsbw  3877  nfifd  4515  nfunid  4878  nfopabd  5179  nfiotadw  6523  nfchnd  18835  nfintd  51187  nfiund  51188