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Theorem rmo0 4361
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 4359 . . 3 ¬ ∃𝑥 ∈ ∅ 𝜑
21pm2.21i 119 . 2 (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑)
3 rmo5 3399 . 2 (∃*𝑥 ∈ ∅ 𝜑 ↔ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑))
42, 3mpbir 231 1 ∃*𝑥 ∈ ∅ 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wrex 3069  ∃!wreu 3377  ∃*wrmo 3378  c0 4332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-dif 3953  df-nul 4333
This theorem is referenced by:  rmosn  4718
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