Theorem nfraldw 3310

Description: Deduction version of nfralw 3312. Version of nfrald 3361 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by NM, 15-Feb-2013.) Avoid ax-9 2153, ax-ext 2735. (Revised by GG, 24-Sep-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 3080	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
2		nfraldw.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfraldw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4	3	nfcrd 2919	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
5		nfraldw.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	4, 5	nfimd 1924	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓))
7	2, 6	nfald 2361	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
8	1, 7	nfxfrd 1884	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  Ⅎwnf 1813   ∈ wcel 2143  Ⅎwnfc 2910  ∀wral 3079

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-10 2176  ax-11 2192  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1810  df-nf 1814  df-clel 2838  df-nfc 2912  df-ral 3080

This theorem is used by:  nfrexdw  3311  nfttrcld  9675  nfchnd  18671