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Theorem reusng 4680
Description: Restricted existential uniqueness over a singleton. (Contributed by AV, 3-Apr-2023.)
Hypothesis
Ref Expression
ralsng.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
reusng (𝐴𝑉 → (∃!𝑥 ∈ {𝐴}𝜑𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem reusng
StepHypRef Expression
1 nfv 1910 . 2 𝑥𝜓
2 ralsng.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2reusngf 4677 1 (𝐴𝑉 → (∃!𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  ∃!wreu 3371  {csn 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-v 3473  df-sbc 3777  df-sn 4630
This theorem is referenced by: (None)
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