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Theorem rmo0 4368
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 4366 . . 3 ¬ ∃𝑥 ∈ ∅ 𝜑
21pm2.21i 119 . 2 (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑)
3 rmo5 3398 . 2 (∃*𝑥 ∈ ∅ 𝜑 ↔ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑))
42, 3mpbir 231 1 ∃*𝑥 ∈ ∅ 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wrex 3068  ∃!wreu 3376  ∃*wrmo 3377  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-dif 3966  df-nul 4340
This theorem is referenced by:  rmosn  4724
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