Theorem nfabdw 2922

Description: Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2923 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-13 2380. (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 1921	. 2 ⊢ Ⅎ𝑧𝜑
2		df-clab 2718	. . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
3		sb6 2096	. . . 4 ⊢ ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓))
4	2, 3	bitri 276	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓))
5		nfabdw.1	. . . 4 ⊢ Ⅎ𝑦𝜑
6		nfvd 1922	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
7		nfabdw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
8	6, 7	nfimd 1901	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = 𝑧 → 𝜓))
9	5, 8	nfald 2337	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 = 𝑧 → 𝜓))
10	4, 9	nfxfrd 1861	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓})
11	1, 10	nfcd 2894	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Syntax hints:   → wi 4  ∀wal 1545  Ⅎwnf 1790  [wsb 2073   ∈ wcel 2119  {cab 2717  Ⅎwnfc 2886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-11 2168  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-nfc 2888

This theorem is referenced by:  nfsbcdw  3744  nfcsb1d  3853  nfcsbw  3857  nfifd  4484  nfunid  4844  nfopabd  5140  nfiotadw  6444  nfchnd  18568  nfintd  50163  nfiund  50164