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Theorem eqv 3418
 Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2142, ax-11 2158, ax-13 2379. (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 2751 . 2 (𝐴 = V ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
2 vex 3413 . . . 4 𝑥 ∈ V
32tbt 373 . . 3 (𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ V))
43albii 1821 . 2 (∀𝑥 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
51, 4bitr4i 281 1 (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111  Vcvv 3409 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411 This theorem is referenced by:  abvALT  3421  dmi  5762  dmep  5764  dfac10  9597  dfac10c  9598  dfac10b  9599  uniwun  10200  fnsingle  33770  bj-abvALT  34628  ttac  40350  nev  40844
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