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Theorem reu2 2804
 Description: A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
reu2 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reu2
StepHypRef Expression
1 nfv 1467 . . 3 𝑦(𝑥𝐴𝜑)
21eu2 1993 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦)))
3 df-reu 2367 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
4 df-rex 2366 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
5 df-ral 2365 . . . 4 (∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
6 19.21v 1802 . . . . . 6 (∀𝑦(𝑥𝐴 → (𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
7 nfv 1467 . . . . . . . . . . . . 13 𝑥 𝑦𝐴
8 nfs1v 1864 . . . . . . . . . . . . 13 𝑥[𝑦 / 𝑥]𝜑
97, 8nfan 1503 . . . . . . . . . . . 12 𝑥(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)
10 eleq1 2151 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
11 sbequ12 1702 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
1210, 11anbi12d 458 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)))
139, 12sbie 1722 . . . . . . . . . . 11 ([𝑦 / 𝑥](𝑥𝐴𝜑) ↔ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
1413anbi2i 446 . . . . . . . . . 10 (((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) ↔ ((𝑥𝐴𝜑) ∧ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)))
15 an4 554 . . . . . . . . . 10 (((𝑥𝐴𝜑) ∧ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
1614, 15bitri 183 . . . . . . . . 9 (((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
1716imbi1i 237 . . . . . . . 8 ((((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦))
18 impexp 260 . . . . . . . 8 ((((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦) ↔ ((𝑥𝐴𝑦𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
19 impexp 260 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑥𝐴 → (𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
2017, 18, 193bitri 205 . . . . . . 7 ((((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
2120albii 1405 . . . . . 6 (∀𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑥𝐴 → (𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
22 df-ral 2365 . . . . . . 7 (∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
2322imbi2i 225 . . . . . 6 ((𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
246, 21, 233bitr4i 211 . . . . 5 (∀𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
2524albii 1405 . . . 4 (∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
265, 25bitr4i 186 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦))
274, 26anbi12i 449 . 2 ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦)))
282, 3, 273bitr4i 211 1 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1288  ∃wex 1427   ∈ wcel 1439  [wsb 1693  ∃!weu 1949  ∀wral 2360  ∃wrex 2361  ∃!wreu 2362 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-eu 1952  df-cleq 2082  df-clel 2085  df-ral 2365  df-rex 2366  df-reu 2367 This theorem is referenced by: (None)
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