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Theorem eqv 3397
Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2185, ax-11 2201, ax-13 2420. (Revised by BJ, 10-Aug-2022.)
Assertion
Ref Expression
eqv (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem eqv
StepHypRef Expression
1 dfcleq 2800 . 2 (𝐴 = V ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
2 vex 3394 . . . 4 𝑥 ∈ V
32tbt 360 . . 3 (𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ V))
43albii 1904 . 2 (∀𝑥 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
51, 4bitr4i 269 1 (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wal 1635   = wceq 1637  wcel 2156  Vcvv 3391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-12 2214  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-tru 1641  df-ex 1860  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-v 3393
This theorem is referenced by:  abv  3400  dmi  5541  dfac10  9244  dfac10c  9245  dfac10b  9246  uniwun  9847  fnsingle  32347  bj-abv  33209  ttac  38104  nev  38562
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