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Theorem eqv 3500
Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2136, ax-11 2151, ax-13 2381. (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 2812 . 2 (𝐴 = V ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
2 vex 3495 . . . 4 𝑥 ∈ V
32tbt 371 . . 3 (𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ V))
43albii 1811 . 2 (∀𝑥 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
51, 4bitr4i 279 1 (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wal 1526   = wceq 1528  wcel 2105  Vcvv 3492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-v 3494
This theorem is referenced by:  abv  3502  dmi  5784  dmep  5786  dfac10  9551  dfac10c  9552  dfac10b  9553  uniwun  10150  fnsingle  33277  bj-abv  34120  ttac  39511  nev  39993
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