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Theorem reu2 3553
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 2009 . . 3 𝑦(𝑥𝐴𝜑)
21eu2 2630 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦)))
3 df-reu 3062 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
4 df-rex 3061 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
5 df-ral 3060 . . . 4 (∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
6 19.21v 2034 . . . . . 6 (∀𝑦(𝑥𝐴 → (𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
7 nfv 2009 . . . . . . . . . . . . 13 𝑥 𝑦𝐴
8 nfs1v 2287 . . . . . . . . . . . . 13 𝑥[𝑦 / 𝑥]𝜑
97, 8nfan 1998 . . . . . . . . . . . 12 𝑥(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)
10 eleq1w 2827 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
11 sbequ12 2278 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
1210, 11anbi12d 624 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)))
139, 12sbie 2499 . . . . . . . . . . 11 ([𝑦 / 𝑥](𝑥𝐴𝜑) ↔ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
1413anbi2i 616 . . . . . . . . . 10 (((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) ↔ ((𝑥𝐴𝜑) ∧ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)))
15 an4 646 . . . . . . . . . 10 (((𝑥𝐴𝜑) ∧ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
1614, 15bitri 266 . . . . . . . . 9 (((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
1716imbi1i 340 . . . . . . . 8 ((((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦))
18 impexp 441 . . . . . . . 8 ((((𝑥𝐴𝑦𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦) ↔ ((𝑥𝐴𝑦𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
19 impexp 441 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑥𝐴 → (𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
2017, 18, 193bitri 288 . . . . . . 7 ((((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
2120albii 1914 . . . . . 6 (∀𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑥𝐴 → (𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
22 df-ral 3060 . . . . . . 7 (∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
2322imbi2i 327 . . . . . 6 ((𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
246, 21, 233bitr4i 294 . . . . 5 (∀𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
2524albii 1914 . . . 4 (∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
265, 25bitr4i 269 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦))
274, 26anbi12i 620 . 2 ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ [𝑦 / 𝑥](𝑥𝐴𝜑)) → 𝑥 = 𝑦)))
282, 3, 273bitr4i 294 1 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650  wex 1874  [wsb 2062  wcel 2155  ∃!weu 2581  wral 3055  wrex 3056  ∃!wreu 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clel 2761  df-ral 3060  df-rex 3061  df-reu 3062
This theorem is referenced by:  reu2eqd  3564  disjinfi  40027
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