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Theorem eqv 3414
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 2151 . 2 (𝐴 = V ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
2 vex 2715 . . . 4 𝑥 ∈ V
32tbt 246 . . 3 (𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ V))
43albii 1450 . 2 (∀𝑥 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
51, 4bitr4i 186 1 (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1333   = wceq 1335  wcel 2128  Vcvv 2712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714
This theorem is referenced by:  setindel  4499  dmi  4803
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