Theorem nfrabwOLD 3319

Description: Obsolete version of nfrabw 3318 as of 23-Nov₂₀₂₄. (Contributed by NM, 13-Oct-2003.) (Revised by Gino Giotto, 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 3073	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2		nftru 1807	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfrabw.2	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
4	3	nfcri 2894	. . . . . 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 1900	. . . 4 ⊢ (⊤ → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
9	2, 8	nfabdw 2930	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
10	9	mptru 1546	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
11	1, 10	nfcxfr 2905	1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 396  ⊤wtru 1540  Ⅎwnf 1786   ∈ wcel 2106  {cab 2715  Ⅎwnfc 2887  {crab 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073

This theorem is referenced by: (None)