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Theorem rmov 3509
Description: An at-most-one quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
rmov (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)

Proof of Theorem rmov
StepHypRef Expression
1 df-rmo 3378 . 2 (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑))
2 vex 3482 . . . 4 𝑥 ∈ V
32biantrur 530 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43mobii 2546 . 2 (∃*𝑥𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑))
51, 4bitr4i 278 1 (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  ∃*wmo 2536  ∃*wrmo 3377  Vcvv 3478
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-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-rmo 3378  df-v 3480
This theorem is referenced by: (None)
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