Theorem nfraldya 2413

Description: Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfraldxy 2411 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)

Hypotheses

Ref	Expression
nfraldya.2	⊢ Ⅎ𝑦𝜑
nfraldya.3	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfraldya.4	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfraldya	⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Distinct variable group:   𝑦,𝐴

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

Proof of Theorem nfraldya

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ral 2365	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
2		sbim 1876	. . . . . 6 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 → 𝜓) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 → [𝑧 / 𝑦]𝜓))
3		clelsb3 2193	. . . . . . 7 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)
4	3	imbi1i 237	. . . . . 6 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐴 → [𝑧 / 𝑦]𝜓) ↔ (𝑧 ∈ 𝐴 → [𝑧 / 𝑦]𝜓))
5	2, 4	bitri 183	. . . . 5 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑧 ∈ 𝐴 → [𝑧 / 𝑦]𝜓))
6	5	albii 1405	. . . 4 ⊢ (∀𝑧[𝑧 / 𝑦](𝑦 ∈ 𝐴 → 𝜓) ↔ ∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑦]𝜓))
7		nfv 1467	. . . . 5 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 → 𝜓)
8	7	sb8 1785	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝜓) ↔ ∀𝑧[𝑧 / 𝑦](𝑦 ∈ 𝐴 → 𝜓))
9		df-ral 2365	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑦]𝜓))
10	6, 8, 9	3bitr4i 211	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝜓) ↔ ∀𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓)
11		nfv 1467	. . . 4 ⊢ Ⅎ𝑧𝜑
12		nfraldya.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
13		nfraldya.2	. . . . 5 ⊢ Ⅎ𝑦𝜑
14		nfraldya.4	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
15	13, 14	nfsbd 1900	. . . 4 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
16	11, 12, 15	nfraldxy 2411	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓)
17	10, 16	nfxfrd 1410	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
18	1, 17	nfxfrd 1410	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1288  Ⅎwnf 1395   ∈ wcel 1439  [wsb 1693  Ⅎwnfc 2216  ∀wral 2360

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365

This theorem is referenced by:  nfralya  2417