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Theorem eqvf 3489
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 2903 . . 3 𝑥V
31, 2cleqf 2932 . 2 (𝐴 = V ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
4 vex 3482 . . . 4 𝑥 ∈ V
54tbt 369 . . 3 (𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ V))
65albii 1816 . 2 (∀𝑥 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
73, 6bitr4i 278 1 (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1535   = wceq 1537  wcel 2106  wnfc 2888  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-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480
This theorem is referenced by: (None)
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