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Theorem rmov 3507
 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 3140 . 2 (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑))
2 vex 3482 . . . 4 𝑥 ∈ V
32biantrur 534 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43mobii 2632 . 2 (∃*𝑥𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑))
51, 4bitr4i 281 1 (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2115  ∃*wmo 2622  ∃*wrmo 3135  Vcvv 3479 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-mo 2624  df-clab 2803  df-cleq 2817  df-clel 2896  df-rmo 3140  df-v 3481 This theorem is referenced by: (None)
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