Theorem nfraldya 2542

Description: Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfraldxy 2540 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 2490	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
2		sbim 1982	. . . . . 6 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 → 𝜓) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 → [𝑧 / 𝑦]𝜓))
3		clelsb1 2311	. . . . . . 7 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)
4	3	imbi1i 238	. . . . . 6 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐴 → [𝑧 / 𝑦]𝜓) ↔ (𝑧 ∈ 𝐴 → [𝑧 / 𝑦]𝜓))
5	2, 4	bitri 184	. . . . 5 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑧 ∈ 𝐴 → [𝑧 / 𝑦]𝜓))
6	5	albii 1494	. . . 4 ⊢ (∀𝑧[𝑧 / 𝑦](𝑦 ∈ 𝐴 → 𝜓) ↔ ∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑦]𝜓))
7		nfv 1552	. . . . 5 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 → 𝜓)
8	7	sb8 1880	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝜓) ↔ ∀𝑧[𝑧 / 𝑦](𝑦 ∈ 𝐴 → 𝜓))
9		df-ral 2490	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑦]𝜓))
10	6, 8, 9	3bitr4i 212	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝜓) ↔ ∀𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓)
11		nfv 1552	. . . 4 ⊢ Ⅎ𝑧𝜑
12		nfraldya.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
13		nfraldya.2	. . . . 5 ⊢ Ⅎ𝑦𝜑
14		nfraldya.4	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
15	13, 14	nfsbd 2006	. . . 4 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
16	11, 12, 15	nfraldxy 2540	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓)
17	10, 16	nfxfrd 1499	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
18	1, 17	nfxfrd 1499	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1371  Ⅎwnf 1484  [wsb 1786   ∈ wcel 2177  Ⅎwnfc 2336  ∀wral 2485

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490

This theorem is referenced by:  nfralya  2547