Theorem nfraldwOLD 3309

Description: Obsolete version of nfraldw 3297 as of 24-Sep-2024. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
nfraldwOLD	⊢ (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 nfraldwOLD

Step	Hyp	Ref	Expression
1		df-ral 3052	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
2		nfraldwOLD.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcvd 2893	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦)
4		nfraldwOLD.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5	3, 4	nfeld 2904	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
6		nfraldwOLD.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
7	5, 6	nfimd 1889	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓))
8	2, 7	nfald 2316	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
9	1, 8	nfxfrd 1848	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1531  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2875  ∀wral 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052

This theorem is referenced by: (None)