Theorem rmov 2793

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 2493	. 2 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 2776	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 303	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	mobii 2092	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 187	1 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃*wmo 2056   ∈ wcel 2177  ∃*wrmo 2488  Vcvv 2773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-rmo 2493  df-v 2775

This theorem is referenced by: (None)