Theorem nfraldw 2540

Description: Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfraldya 2543 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by NM, 15-Feb-2013.) (Revised by GG, 10-Jan-2024.)

Hypotheses

Ref	Expression
nfraldw.1	⊢ Ⅎ𝑦𝜑
nfraldw.2	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfraldw.3	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

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

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem nfraldw

Step	Hyp	Ref	Expression
1		df-ral 2491	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
2		nfraldw.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcvd 2351	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦)
4		nfraldw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5	3, 4	nfeld 2366	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
6		nfraldw.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
7	5, 6	nfimd 1609	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓))
8	2, 7	nfald 1784	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
9	1, 8	nfxfrd 1499	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1371  Ⅎwnf 1484   ∈ wcel 2178  Ⅎwnfc 2337  ∀wral 2486

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491

This theorem is referenced by:  nfralw  2545