Theorem rmov 3487

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 3372	. 2 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 3462	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 540	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	mobii 2579	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 281	1 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  ∃*wmo 2568  ∃*wrmo 3371  Vcvv 3458

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-mo 2570  df-clab 2745  df-cleq 2758  df-clel 2841  df-rmo 3372  df-v 3460

This theorem is used by: (None)