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Theorem rexreusng 4580
Description: Restricted existential uniqueness over a singleton is equivalent to a restricted existential quantification over a singleton. (Contributed by AV, 3-Apr-2023.)
Assertion
Ref Expression
rexreusng (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃!𝑥 ∈ {𝐴}𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem rexreusng
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2802 . . . . 5 (([𝐴 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜑) → 𝐴 = 𝐴)
2 nfsbc1v 3743 . . . . . . . 8 𝑦[𝐴 / 𝑦][𝐴 / 𝑥]𝜑
3 nfv 1915 . . . . . . . 8 𝑦[𝐴 / 𝑥]𝜑
42, 3nfan 1900 . . . . . . 7 𝑦([𝐴 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
5 nfv 1915 . . . . . . 7 𝑦 𝐴 = 𝐴
64, 5nfim 1897 . . . . . 6 𝑦(([𝐴 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜑) → 𝐴 = 𝐴)
7 sbceq1a 3734 . . . . . . . 8 (𝑦 = 𝐴 → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦][𝐴 / 𝑥]𝜑))
8 dfsbcq2 3726 . . . . . . . 8 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
97, 8anbi12d 633 . . . . . . 7 (𝑦 = 𝐴 → (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ ([𝐴 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)))
10 eqeq2 2813 . . . . . . 7 (𝑦 = 𝐴 → (𝐴 = 𝑦𝐴 = 𝐴))
119, 10imbi12d 348 . . . . . 6 (𝑦 = 𝐴 → ((([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦) ↔ (([𝐴 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜑) → 𝐴 = 𝐴)))
126, 11ralsngf 4574 . . . . 5 (𝐴𝑉 → (∀𝑦 ∈ {𝐴} (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦) ↔ (([𝐴 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜑) → 𝐴 = 𝐴)))
131, 12mpbiri 261 . . . 4 (𝐴𝑉 → ∀𝑦 ∈ {𝐴} (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦))
14 nfcv 2958 . . . . . 6 𝑥{𝐴}
15 nfsbc1v 3743 . . . . . . . 8 𝑥[𝐴 / 𝑥]𝜑
16 nfs1v 2158 . . . . . . . 8 𝑥[𝑦 / 𝑥]𝜑
1715, 16nfan 1900 . . . . . . 7 𝑥([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑)
18 nfv 1915 . . . . . . 7 𝑥 𝐴 = 𝑦
1917, 18nfim 1897 . . . . . 6 𝑥(([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦)
2014, 19nfralw 3192 . . . . 5 𝑥𝑦 ∈ {𝐴} (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦)
21 sbceq1a 3734 . . . . . . . 8 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
2221anbi1d 632 . . . . . . 7 (𝑥 = 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑)))
23 eqeq1 2805 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
2422, 23imbi12d 348 . . . . . 6 (𝑥 = 𝐴 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦)))
2524ralbidv 3165 . . . . 5 (𝑥 = 𝐴 → (∀𝑦 ∈ {𝐴} ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {𝐴} (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦)))
2620, 25ralsngf 4574 . . . 4 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {𝐴} (([𝐴 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝐴 = 𝑦)))
2713, 26mpbird 260 . . 3 (𝐴𝑉 → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
2827biantrud 535 . 2 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
29 reu2 3667 . 2 (∃!𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
3028, 29syl6bbr 292 1 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃!𝑥 ∈ {𝐴}𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  [wsb 2069  wcel 2112  wral 3109  wrex 3110  ∃!wreu 3111  [wsbc 3723  {csn 4528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-reu 3116  df-v 3446  df-sbc 3724  df-sn 4529
This theorem is referenced by: (None)
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