Theorem rexv 2781

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 2481	. 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 2766	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 303	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	exbii 1619	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 187	1 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1506   ∈ wcel 2167  ∃wrex 2476  Vcvv 2763

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-rex 2481  df-v 2765

This theorem is referenced by:  rexcom4  2786  spesbc  3075  abnex  4482  dfco2  5169  dfco2a  5170