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Theorem rmoeq 3566
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 3535 . . 3 ∃*𝑥 𝑥 = 𝐴
21moani 2647 . 2 ∃*𝑥(𝑥𝐵𝑥 = 𝐴)
3 df-rmo 3063 . 2 (∃*𝑥𝐵 𝑥 = 𝐴 ↔ ∃*𝑥(𝑥𝐵𝑥 = 𝐴))
42, 3mpbir 222 1 ∃*𝑥𝐵 𝑥 = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  wcel 2155  ∃*wmo 2563  ∃*wrmo 3058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-cleq 2758  df-rmo 3063
This theorem is referenced by:  nbusgredgeu  26546
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