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Theorem reusng 4641
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 1918 . 2 𝑥𝜓
2 ralsng.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2reusngf 4638 1 (𝐴𝑉 → (∃!𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  ∃!wreu 3354  {csn 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-v 3450  df-sbc 3745  df-sn 4592
This theorem is referenced by: (None)
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