Theorem nfreuwOLD 3421

Description: Obsolete version of nfreuw 3409 as of 21-Nov-2024. (Contributed by NM, 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
nfreuwOLD	⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem nfreuwOLD

Step	Hyp	Ref	Expression
1		df-reu 3376	. . 3 ⊢ (∃!𝑦 ∈ 𝐴 𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nftru 1805	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfcvd 2903	. . . . . 6 ⊢ (⊤ → Ⅎ𝑥𝑦)
4		nfreuwOLD.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
5	4	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑥𝐴)
6	3, 5	nfeld 2913	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfreuwOLD.2	. . . . . 6 ⊢ Ⅎ𝑥𝜑
8	7	a1i 11	. . . . 5 ⊢ (⊤ → Ⅎ𝑥𝜑)
9	6, 8	nfand 1899	. . . 4 ⊢ (⊤ → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
10	2, 9	nfeudw 2584	. . 3 ⊢ (⊤ → Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
11	1, 10	nfxfrd 1855	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑)
12	11	mptru 1547	1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 395  ⊤wtru 1541  Ⅎwnf 1784   ∈ wcel 2105  ∃!weu 2561  Ⅎwnfc 2882  ∃!wreu 3373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-mo 2533  df-eu 2562  df-cleq 2723  df-clel 2809  df-nfc 2884  df-reu 3376

This theorem is referenced by: (None)