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Theorem eqv 3195
Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.)
Assertion
Ref Expression
eqv (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem eqv
StepHypRef Expression
1 nfcv 2761 . 2 𝑥𝐴
21eqvf 3194 1 (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1478   = wceq 1480  wcel 1987  Vcvv 3190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192
This theorem is referenced by:  abv  3196  dmi  5310  dfac10  8919  dfac10c  8920  dfac10b  8921  uniwun  9522  fnsingle  31721  bj-abv  32601  ttac  37122  nev  37582
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