Theorem nfabdw 2318

Description: Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2319 with a disjoint variable condition. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by Gino Giotto, 10-Jan-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 1508	. 2 ⊢ Ⅎ𝑧𝜑
2		df-clab 2144	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
3		nfabdw.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfabdw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	3, 4	alrimi 1502	. . . 4 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥𝜓)
6		nfa1 1521	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦Ⅎ𝑥𝜓
7		sb6 1866	. . . . . . . . . . . 12 ⊢ ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓))
8	7	a1i 9	. . . . . . . . . . 11 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓)))
9	7	biimpri 132	. . . . . . . . . . . 12 ⊢ (∀𝑦(𝑦 = 𝑧 → 𝜓) → [𝑧 / 𝑦]𝜓)
10	9	axc4i 1522	. . . . . . . . . . 11 ⊢ (∀𝑦(𝑦 = 𝑧 → 𝜓) → ∀𝑦[𝑧 / 𝑦]𝜓)
11	8, 10	syl6bi 162	. . . . . . . . . 10 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ([𝑧 / 𝑦]𝜓 → ∀𝑦[𝑧 / 𝑦]𝜓))
12	6, 11	nf5d 2005	. . . . . . . . 9 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑦[𝑧 / 𝑦]𝜓)
13	6, 12	nfim1 1551	. . . . . . . 8 ⊢ Ⅎ𝑦(∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓)
14		sbequ12 1751	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝜓 ↔ [𝑧 / 𝑦]𝜓))
15	14	imbi2d 229	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((∀𝑦Ⅎ𝑥𝜓 → 𝜓) ↔ (∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓)))
16	13, 15	equsalv 1773	. . . . . . 7 ⊢ (∀𝑦(𝑦 = 𝑧 → (∀𝑦Ⅎ𝑥𝜓 → 𝜓)) ↔ (∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓))
17	16	bicomi 131	. . . . . 6 ⊢ ((∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ ∀𝑦(𝑦 = 𝑧 → (∀𝑦Ⅎ𝑥𝜓 → 𝜓)))
18		nfv 1508	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = 𝑧
19		nfnf1 1524	. . . . . . . . . 10 ⊢ Ⅎ𝑥Ⅎ𝑥𝜓
20	19	nfal 1556	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜓
21		sp 1491	. . . . . . . . 9 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓)
22	20, 21	nfim1 1551	. . . . . . . 8 ⊢ Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜓 → 𝜓)
23	18, 22	nfim 1552	. . . . . . 7 ⊢ Ⅎ𝑥(𝑦 = 𝑧 → (∀𝑦Ⅎ𝑥𝜓 → 𝜓))
24	23	nfal 1556	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦(𝑦 = 𝑧 → (∀𝑦Ⅎ𝑥𝜓 → 𝜓))
25	17, 24	nfxfr 1454	. . . . 5 ⊢ Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓)
26		pm5.5 241	. . . . . 6 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ((∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ [𝑧 / 𝑦]𝜓))
27	20, 26	nfbidf 1519	. . . . 5 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜓))
28	25, 27	mpbii 147	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
29	5, 28	syl 14	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
30	2, 29	nfxfrd 1455	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓})
31	1, 30	nfcd 2294	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1333  Ⅎwnf 1440  [wsb 1742   ∈ wcel 2128  {cab 2143  Ⅎwnfc 2286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-nfc 2288

This theorem is referenced by:  nfsbcdw  3065  nfcsbw  3067