Theorem rmov 3510

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 3379	. 2 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 3483	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 530	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	mobii 2547	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 278	1 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2107  ∃*wmo 2537  ∃*wrmo 3378  Vcvv 3479

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-ex 1779  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-rmo 3379  df-v 3481

This theorem is referenced by: (None)