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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
StepHypRef Expression
1 dfcleq 2347 . 2 (A = V ↔ x(x Ax V))
2 vex 2862 . . . 4 x V
32tbt 333 . . 3 (x A ↔ (x Ax V))
43albii 1566 . 2 (x x Ax(x Ax V))
51, 4bitr4i 243 1 (A = V ↔ x x A)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2859 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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
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