Theorem nfreuw 3373

Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3375 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

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

Assertion

Ref	Expression
nfreuw	⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem nfreuw

Step	Hyp	Ref	Expression
1		df-reu 3144	. . 3 ⊢ (∃!𝑦 ∈ 𝐴 𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nftru 1804	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfcvd 2977	. . . . . 6 ⊢ (⊤ → Ⅎ𝑥𝑦)
4		nfreuw.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
5	4	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑥𝐴)
6	3, 5	nfeld 2988	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfreuw.2	. . . . . 6 ⊢ Ⅎ𝑥𝜑
8	7	a1i 11	. . . . 5 ⊢ (⊤ → Ⅎ𝑥𝜑)
9	6, 8	nfand 1897	. . . 4 ⊢ (⊤ → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
10	2, 9	nfeudw 2676	. . 3 ⊢ (⊤ → Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
11	1, 10	nfxfrd 1853	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑)
12	11	mptru 1543	1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 398  ⊤wtru 1537  Ⅎwnf 1783   ∈ wcel 2113  ∃!weu 2652  Ⅎwnfc 2960  ∃!wreu 3139

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-mo 2621  df-eu 2653  df-cleq 2813  df-clel 2892  df-nfc 2962  df-reu 3144

This theorem is referenced by:  sbcreu  3853  reuccatpfxs1  14104  2reu7  43384  2reu8  43385