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Theorem rmoeq 3555
 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 3532 . . 3 ∃*𝑥 𝑥 = 𝐴
21moani 2673 . 2 ∃*𝑥(𝑥𝐵𝑥 = 𝐴)
3 df-rmo 3068 . 2 (∃*𝑥𝐵 𝑥 = 𝐴 ↔ ∃*𝑥(𝑥𝐵𝑥 = 𝐴))
42, 3mpbir 221 1 ∃*𝑥𝐵 𝑥 = 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 382   = wceq 1630   ∈ wcel 2144  ∃*wmo 2618  ∃*wrmo 3063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-rmo 3068  df-v 3351 This theorem is referenced by:  nbusgredgeu  26489
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