Theorem nfreudxy 2532

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 2223	. . . . . 6 ⊢ Ⅎ𝑥𝑦
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦)
4		nfreudxy.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5	3, 4	nfeld 2238	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
6		nfreudxy.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
7	5, 6	nfand 1501	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
8	1, 7	nfeud 1959	. 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
9		df-reu 2360	. . 3 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
10	9	nfbii 1403	. 2 ⊢ (Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓 ↔ Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
11	8, 10	sylibr 132	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 102  Ⅎwnf 1390   ∈ wcel 1434  ∃!weu 1943  Ⅎwnfc 2210  ∃!wreu 2355

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-cleq 2076  df-clel 2079  df-nfc 2212  df-reu 2360

This theorem is referenced by:  nfreuxy  2533