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Theorem eqv 3300
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 dfcleq 2082 . 2 (𝐴 = V ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
2 vex 2622 . . . 4 𝑥 ∈ V
32tbt 245 . . 3 (𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ V))
43albii 1404 . 2 (∀𝑥 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
51, 4bitr4i 185 1 (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  wb 103  wal 1287   = wceq 1289  wcel 1438  Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  setindel  4344  dmi  4639
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