Theorem eqv 3234

Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.)

Assertion

Ref	Expression
eqv	⊢ (A = V ↔ ∀x x ∈ A)

Distinct variable group:   x,A

Proof of Theorem eqv

Step	Hyp	Ref	Expression
1		dfcleq 2031	. 2 ⊢ (A = V ↔ ∀x(x ∈ A ↔ x ∈ V))
2		vex 2554	. . . 4 ⊢ x ∈ V
3	2	tbt 236	. . 3 ⊢ (x ∈ A ↔ (x ∈ A ↔ x ∈ V))
4	3	albii 1356	. 2 ⊢ (∀x x ∈ A ↔ ∀x(x ∈ A ↔ x ∈ V))
5	1, 4	bitr4i 176	1 ⊢ (A = V ↔ ∀x x ∈ A)

Syntax hints:   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  Vcvv 2551

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  setindel  4221  dmi  4493