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Theorem reu2 3361
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 1830 . . 3 𝑦(𝑥𝐴𝜑)
21eu2 2497 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦)))
3 df-reu 2903 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
4 df-rex 2902 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
5 df-ral 2901 . . . 4 (∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
6 19.21v 1855 . . . . . 6 (∀𝑦(𝑥𝐴 → (𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
7 nfv 1830 . . . . . . . . . . . . 13 𝑥 𝑦𝐴
8 nfs1v 2425 . . . . . . . . . . . . 13 𝑥[𝑦 / 𝑥]𝜑
97, 8nfan 1816 . . . . . . . . . . . 12 𝑥(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)
10 eleq1 2676 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
11 sbequ12 2097 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
1210, 11anbi12d 743 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)))
139, 12sbie 2396 . . . . . . . . . . 11 ([𝑦 / 𝑥](𝑥𝐴𝜑) ↔ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
1413anbi2i 726 . . . . . . . . . 10 (((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) ↔ ((𝑥𝐴𝜑) ∧ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)))
15 an4 861 . . . . . . . . . 10 (((𝑥𝐴𝜑) ∧ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
1614, 15bitri 263 . . . . . . . . 9 (((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
1716imbi1i 338 . . . . . . . 8 ((((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦))
18 impexp 461 . . . . . . . 8 ((((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦) ↔ ((𝑥𝐴𝑦𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
19 impexp 461 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑥𝐴 → (𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
2017, 18, 193bitri 285 . . . . . . 7 ((((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
2120albii 1737 . . . . . 6 (∀𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑥𝐴 → (𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
22 df-ral 2901 . . . . . . 7 (∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
2322imbi2i 325 . . . . . 6 ((𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
246, 21, 233bitr4i 291 . . . . 5 (∀𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
2524albii 1737 . . . 4 (∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
265, 25bitr4i 266 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦))
274, 26anbi12i 729 . 2 ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦)))
282, 3, 273bitr4i 291 1 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wal 1473  wex 1695  [wsb 1867  wcel 1977  ∃!weu 2458  wral 2896  wrex 2897  ∃!wreu 2898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-cleq 2603  df-clel 2606  df-ral 2901  df-rex 2902  df-reu 2903
This theorem is referenced by:  reu2eqd  3370  disjinfi  38375
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