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Theorem eqvf 3397
Description: The universe contains every set. (Contributed by BJ, 15-Jul-2021.)
Hypothesis
Ref Expression
eqvf.1 𝑥𝐴
Assertion
Ref Expression
eqvf (𝐴 = V ↔ ∀𝑥 𝑥𝐴)

Proof of Theorem eqvf
StepHypRef Expression
1 eqvf.1 . . 3 𝑥𝐴
2 nfcv 2947 . . 3 𝑥V
31, 2cleqf 2973 . 2 (𝐴 = V ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
4 vex 3393 . . . 4 𝑥 ∈ V
54tbt 360 . . 3 (𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ V))
65albii 1907 . 2 (∀𝑥 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
73, 6bitr4i 269 1 (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wal 1635   = wceq 1637  wcel 2158  wnfc 2934  Vcvv 3390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-v 3392
This theorem is referenced by:  eqvOLD  3398
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