Theorem rexv 1824

Description: An existential quantifier restricted to the universe is unrestricted.

Assertion

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

Proof of Theorem rexv

Step	Hyp	Ref	Expression
1		df-rex 1653	. 2 ⊢ (∃x ∈ V φ ↔ ∃x(x ∈ V ⋀ φ))
2		visset 1816	. . . 4 ⊢ x ∈ V
3	2	biantrur 727	. . 3 ⊢ (φ ↔ (x ∈ V ⋀ φ))
4	3	exbii 1053	. 2 ⊢ (∃xφ ↔ ∃x(x ∈ V ⋀ φ))
5	1, 4	bitr4 176	1 ⊢ (∃x ∈ V φ ↔ ∃xφ)

Syntax hints:   ↔ wb 146   ⋀ wa 223   ∈ wcel 960  ∃wex 982  ∃wrex 1649  Vcvv 1814

This theorem is referenced by:  rexcom4 1827  ac6s2 4768

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rex 1653  df-v 1815

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