Theorem nfabdw 2351

Description: Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2352 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 1539	. 2 ⊢ Ⅎ𝑧𝜑
2		df-clab 2176	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
3		nfabdw.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfabdw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	3, 4	alrimi 1533	. . . 4 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥𝜓)
6		nfa1 1552	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦Ⅎ𝑥𝜓
7		sb6 1898	. . . . . . . . . . . 12 ⊢ ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓))
8	7	a1i 9	. . . . . . . . . . 11 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓)))
9	7	biimpri 133	. . . . . . . . . . . 12 ⊢ (∀𝑦(𝑦 = 𝑧 → 𝜓) → [𝑧 / 𝑦]𝜓)
10	9	axc4i 1553	. . . . . . . . . . 11 ⊢ (∀𝑦(𝑦 = 𝑧 → 𝜓) → ∀𝑦[𝑧 / 𝑦]𝜓)
11	8, 10	biimtrdi 163	. . . . . . . . . 10 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ([𝑧 / 𝑦]𝜓 → ∀𝑦[𝑧 / 𝑦]𝜓))
12	6, 11	nf5d 2037	. . . . . . . . 9 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑦[𝑧 / 𝑦]𝜓)
13	6, 12	nfim1 1582	. . . . . . . 8 ⊢ Ⅎ𝑦(∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓)
14		sbequ12 1782	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝜓 ↔ [𝑧 / 𝑦]𝜓))
15	14	imbi2d 230	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((∀𝑦Ⅎ𝑥𝜓 → 𝜓) ↔ (∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓)))
16	13, 15	equsalv 1804	. . . . . . 7 ⊢ (∀𝑦(𝑦 = 𝑧 → (∀𝑦Ⅎ𝑥𝜓 → 𝜓)) ↔ (∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓))
17	16	bicomi 132	. . . . . 6 ⊢ ((∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ ∀𝑦(𝑦 = 𝑧 → (∀𝑦Ⅎ𝑥𝜓 → 𝜓)))
18		nfv 1539	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = 𝑧
19		nfnf1 1555	. . . . . . . . . 10 ⊢ Ⅎ𝑥Ⅎ𝑥𝜓
20	19	nfal 1587	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜓
21		sp 1522	. . . . . . . . 9 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓)
22	20, 21	nfim1 1582	. . . . . . . 8 ⊢ Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜓 → 𝜓)
23	18, 22	nfim 1583	. . . . . . 7 ⊢ Ⅎ𝑥(𝑦 = 𝑧 → (∀𝑦Ⅎ𝑥𝜓 → 𝜓))
24	23	nfal 1587	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦(𝑦 = 𝑧 → (∀𝑦Ⅎ𝑥𝜓 → 𝜓))
25	17, 24	nfxfr 1485	. . . . 5 ⊢ Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓)
26		pm5.5 242	. . . . . 6 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ((∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ [𝑧 / 𝑦]𝜓))
27	20, 26	nfbidf 1550	. . . . 5 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜓))
28	25, 27	mpbii 148	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
29	5, 28	syl 14	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
30	2, 29	nfxfrd 1486	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓})
31	1, 30	nfcd 2327	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362  Ⅎwnf 1471  [wsb 1773   ∈ wcel 2160  {cab 2175  Ⅎwnfc 2319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-nfc 2321

This theorem is referenced by:  nfsbcdw  3106  nfcsbw  3108