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Theorem rmoeq 3635
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 3604 . . 3 ∃*𝑥 𝑥 = 𝐴
21moani 2553 . 2 ∃*𝑥(𝑥𝐵𝑥 = 𝐴)
3 df-rmo 3061 . 2 (∃*𝑥𝐵 𝑥 = 𝐴 ↔ ∃*𝑥(𝑥𝐵𝑥 = 𝐴))
42, 3mpbir 234 1 ∃*𝑥𝐵 𝑥 = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1542  wcel 2113  ∃*wmo 2538  ∃*wrmo 3056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-9 2123  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-mo 2540  df-cleq 2730  df-rmo 3061
This theorem is referenced by:  nbusgredgeu  27300
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