Theorem nfabdwOLD 2921

Description: Obsolete version of nfabdw 2920 as of 23-Sep-2024. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfabdw.1	⊢ Ⅎ𝑦𝜑
nfabdw.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfabdwOLD	⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem nfabdwOLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1922	. 2 ⊢ Ⅎ𝑧𝜑
2		df-clab 2715	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
3		nfabdw.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfabdw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	3, 4	alrimi 2213	. . . 4 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥𝜓)
6		nfa1 2154	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦Ⅎ𝑥𝜓
7		sb6 2093	. . . . . . . . . . . 12 ⊢ ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓))
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓)))
9	7	biimpri 231	. . . . . . . . . . . 12 ⊢ (∀𝑦(𝑦 = 𝑧 → 𝜓) → [𝑧 / 𝑦]𝜓)
10	9	axc4i 2323	. . . . . . . . . . 11 ⊢ (∀𝑦(𝑦 = 𝑧 → 𝜓) → ∀𝑦[𝑧 / 𝑦]𝜓)
11	8, 10	syl6bi 256	. . . . . . . . . 10 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ([𝑧 / 𝑦]𝜓 → ∀𝑦[𝑧 / 𝑦]𝜓))
12	6, 11	nf5d 2287	. . . . . . . . 9 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑦[𝑧 / 𝑦]𝜓)
13	6, 12	nfim1 2199	. . . . . . . 8 ⊢ Ⅎ𝑦(∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓)
14		sbequ12 2251	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝜓 ↔ [𝑧 / 𝑦]𝜓))
15	14	imbi2d 344	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((∀𝑦Ⅎ𝑥𝜓 → 𝜓) ↔ (∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓)))
16	13, 15	equsalv 2266	. . . . . . 7 ⊢ (∀𝑦(𝑦 = 𝑧 → (∀𝑦Ⅎ𝑥𝜓 → 𝜓)) ↔ (∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓))
17	16	bicomi 227	. . . . . 6 ⊢ ((∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ ∀𝑦(𝑦 = 𝑧 → (∀𝑦Ⅎ𝑥𝜓 → 𝜓)))
18		nfv 1922	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = 𝑧
19		nfnf1 2157	. . . . . . . . . 10 ⊢ Ⅎ𝑥Ⅎ𝑥𝜓
20	19	nfal 2324	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜓
21		sp 2182	. . . . . . . . 9 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓)
22	20, 21	nfim1 2199	. . . . . . . 8 ⊢ Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜓 → 𝜓)
23	18, 22	nfim 1904	. . . . . . 7 ⊢ Ⅎ𝑥(𝑦 = 𝑧 → (∀𝑦Ⅎ𝑥𝜓 → 𝜓))
24	23	nfal 2324	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦(𝑦 = 𝑧 → (∀𝑦Ⅎ𝑥𝜓 → 𝜓))
25	17, 24	nfxfr 1860	. . . . 5 ⊢ Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓)
26		pm5.5 365	. . . . . 6 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ((∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ [𝑧 / 𝑦]𝜓))
27	20, 26	nfbidf 2224	. . . . 5 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜓))
28	25, 27	mpbii 236	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
29	5, 28	syl 17	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
30	2, 29	nfxfrd 1861	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓})
31	1, 30	nfcd 2885	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541  Ⅎwnf 1791  [wsb 2072   ∈ wcel 2112  {cab 2714  Ⅎwnfc 2877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-10 2143  ax-11 2160  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-nfc 2879

This theorem is referenced by: (None)