Theorem rmov 2706

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 2424	. 2 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 2689	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 301	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	mobii 2036	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 186	1 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)

Syntax hints:   ∧ wa 103   ↔ wb 104   ∈ wcel 1480  ∃*wmo 2000  ∃*wrmo 2419  Vcvv 2686

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-rmo 2424  df-v 2688

This theorem is referenced by: (None)