Theorem rmov 2750

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

Step	Hyp	Ref	Expression
1		df-rmo 2456	. 2 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 2733	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 301	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	mobii 2056	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 186	1 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃*wmo 2020   ∈ wcel 2141  ∃*wrmo 2451  Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-rmo 2456  df-v 2732

This theorem is referenced by: (None)