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Theorem rmoeq 3372
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 3349 . . 3 ∃*𝑥 𝑥 = 𝐴
21moani 2513 . 2 ∃*𝑥(𝑥𝐵𝑥 = 𝐴)
3 df-rmo 2904 . 2 (∃*𝑥𝐵 𝑥 = 𝐴 ↔ ∃*𝑥(𝑥𝐵𝑥 = 𝐴))
42, 3mpbir 220 1 ∃*𝑥𝐵 𝑥 = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1475  wcel 1977  ∃*wmo 2459  ∃*wrmo 2899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-rmo 2904  df-v 3175
This theorem is referenced by:  nbusgredgeu  40594
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