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Theorem rmoeq 3438
 Description: Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.) (Revised by AV, 27-Oct-2020.) (Proof shortened by NM, 29-Oct-2020.)
Assertion
Ref Expression
rmoeq ∃*𝑥𝐵 𝑥 = 𝐴
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem rmoeq
StepHypRef Expression
1 moeq 3415 . . 3 ∃*𝑥 𝑥 = 𝐴
21moani 2554 . 2 ∃*𝑥(𝑥𝐵𝑥 = 𝐴)
3 df-rmo 2949 . 2 (∃*𝑥𝐵 𝑥 = 𝐴 ↔ ∃*𝑥(𝑥𝐵𝑥 = 𝐴))
42, 3mpbir 221 1 ∃*𝑥𝐵 𝑥 = 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∃*wmo 2499  ∃*wrmo 2944 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-rmo 2949  df-v 3233 This theorem is referenced by:  nbusgredgeu  26312
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