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Theorem eqv 3490
Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2141, ax-11 2157, ax-13 2377. (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 2730 . 2 (𝐴 = V ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
2 vex 3484 . . . 4 𝑥 ∈ V
32tbt 369 . . 3 (𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ V))
43albii 1819 . 2 (∀𝑥 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
51, 4bitr4i 278 1 (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1538   = wceq 1540  wcel 2108  Vcvv 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482
This theorem is referenced by:  abvALT  3493  dmi  5932  dmep  5934  dfac10  10178  dfac10c  10179  dfac10b  10180  uniwun  10780  fnsingle  35920  bj-abvALT  36908  ttac  43048  nev  43783
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