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Theorem eqv 3488
Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2139, ax-11 2155, ax-13 2375. (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 2728 . 2 (𝐴 = V ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
2 vex 3482 . . . 4 𝑥 ∈ V
32tbt 369 . . 3 (𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ V))
43albii 1816 . 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 1535   = wceq 1537  wcel 2106  Vcvv 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480
This theorem is referenced by:  abvALT  3491  dmi  5935  dmep  5937  dfac10  10176  dfac10c  10177  dfac10b  10178  uniwun  10778  fnsingle  35901  bj-abvALT  36890  ttac  43025  nev  43760
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