Theorem eqv 3565

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 2347	. 2 ⊢ (A = V ↔ ∀x(x ∈ A ↔ x ∈ V))
2		vex 2862	. . . 4 ⊢ x ∈ V
3	2	tbt 333	. . 3 ⊢ (x ∈ A ↔ (x ∈ A ↔ x ∈ V))
4	3	albii 1566	. 2 ⊢ (∀x x ∈ A ↔ ∀x(x ∈ A ↔ x ∈ V))
5	1, 4	bitr4i 243	1 ⊢ (A = V ↔ ∀x x ∈ A)

Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861

This theorem is referenced by:  nincompl  4072  xpvv  4843  dmi  4919  1stfo  5505  2ndfo  5506  swapf1o  5511  dmep  5524