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Theorem rmo0 4316
Description: Vacuous restricted at-most-one quantifier is always true. (Contributed by AV, 3-Apr-2023.)
Assertion
Ref Expression
rmo0 ∃*𝑥 ∈ ∅ 𝜑

Proof of Theorem rmo0
StepHypRef Expression
1 rex0 4314 . . 3 ¬ ∃𝑥 ∈ ∅ 𝜑
21pm2.21i 119 . 2 (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑)
3 rmo5 3432 . 2 (∃*𝑥 ∈ ∅ 𝜑 ↔ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑))
42, 3mpbir 232 1 ∃*𝑥 ∈ ∅ 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wrex 3136  ∃!wreu 3137  ∃*wrmo 3138  c0 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-dif 3936  df-nul 4289
This theorem is referenced by:  rmosn  4647
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