Theorem rexv 2651

Description: An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Assertion

Ref	Expression
rexv	⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)

Proof of Theorem rexv

Step	Hyp	Ref	Expression
1		df-rex 2376	. 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 2636	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 298	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	exbii 1548	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 186	1 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1433   ∈ wcel 1445  ∃wrex 2371  Vcvv 2633

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-rex 2376  df-v 2635

This theorem is referenced by:  rexcom4  2656  spesbc  2938  abnex  4297  dfco2  4964  dfco2a  4965