Theorem nfreudxy 2682

Description: Not-free deduction for restricted uniqueness. This is a version where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)

Hypotheses

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

Assertion

Ref	Expression
nfreudxy	⊢ (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 nfreudxy

Step	Hyp	Ref	Expression
1		nfreudxy.1	. . 3 ⊢ Ⅎ𝑦𝜑
2		nfcv 2350	. . . . . 6 ⊢ Ⅎ𝑥𝑦
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦)
4		nfreudxy.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5	3, 4	nfeld 2366	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
6		nfreudxy.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
7	5, 6	nfand 1592	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
8	1, 7	nfeud 2071	. 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
9		df-reu 2493	. . 3 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
10	9	nfbii 1497	. 2 ⊢ (Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓 ↔ Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
11	8, 10	sylibr 134	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 104  Ⅎwnf 1484  ∃!weu 2055   ∈ wcel 2178  Ⅎwnfc 2337  ∃!wreu 2488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-cleq 2200  df-clel 2203  df-nfc 2339  df-reu 2493

This theorem is referenced by:  nfreuw  2683