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Theorem rmo0 4352
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 4350 . . 3 ¬ ∃𝑥 ∈ ∅ 𝜑
21pm2.21i 119 . 2 (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑)
3 rmo5 3388 . 2 (∃*𝑥 ∈ ∅ 𝜑 ↔ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑))
42, 3mpbir 230 1 ∃*𝑥 ∈ ∅ 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wrex 3062  ∃!wreu 3366  ∃*wrmo 3367  c0 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-dif 3944  df-nul 4316
This theorem is referenced by:  rmosn  4716
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