Theorem nfrabwOLD 3460

Description: Obsolete version of nfrabw 3459 as of 23-Nov-2024. (Contributed by NM, 13-Oct-2003.) (Revised by GG, 10-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
nfrabw.1	⊢ Ⅎ𝑥𝜑
nfrabw.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfrabwOLD	⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem nfrabwOLD

Step	Hyp	Ref	Expression
1		df-rab 3421	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2		nftru 1804	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfrabw.2	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
4	3	nfcri 2891	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
5	4	a1i 11	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑦 ∈ 𝐴)
6		nfrabw.1	. . . . . 6 ⊢ Ⅎ𝑥𝜑
7	6	a1i 11	. . . . 5 ⊢ (⊤ → Ⅎ𝑥𝜑)
8	5, 7	nfand 1897	. . . 4 ⊢ (⊤ → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
9	2, 8	nfabdw 2921	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
10	9	mptru 1547	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
11	1, 10	nfcxfr 2897	1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 395  ⊤wtru 1541  Ⅎwnf 1783   ∈ wcel 2109  {cab 2714  Ⅎwnfc 2884  {crab 3420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-rab 3421

This theorem is referenced by: (None)