Theorem nfabdw 2943

Description: Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2944 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-13 2401. (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 1946	. 2 ⊢ Ⅎ𝑧𝜑
2		df-clab 2739	. . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
3		sb6 2121	. . . 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 1947	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
7		nfabdw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
8	6, 7	nfimd 1926	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = 𝑧 → 𝜓))
9	5, 8	nfald 2358	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 = 𝑧 → 𝜓))
10	4, 9	nfxfrd 1886	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓})
11	1, 10	nfcd 2915	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  Ⅎwnf 1815  [wsb 2098   ∈ wcel 2144  {cab 2738  Ⅎwnfc 2907

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-nfc 2909

This theorem is used by:  nfsbcdw  3759  nfcsb1d  3869  nfcsbw  3873  nfifd  4511  nfunid  4872  nfopabd  5172  nfiotadw  6500  nfchnd  18794  nfintd  50992  nfiund  50993