Theorem nfreudxy 2725

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 2392	. . . . . 6 ⊢ Ⅎ𝑥𝑦
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦)
4		nfreudxy.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5	3, 4	nfeld 2408	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
6		nfreudxy.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
7	5, 6	nfand 1621	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
8	1, 7	nfeud 2102	. 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
9		df-reu 2535	. . 3 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
10	9	nfbii 1526	. 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 1513  ∃!weu 2086   ∈ wcel 2209  Ⅎwnfc 2379  ∃!wreu 2530

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-cleq 2231  df-clel 2234  df-nfc 2381  df-reu 2535

This theorem is referenced by:  nfreuw  2726