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Theorem eqv 3216
Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.)
Assertion
Ref Expression
eqv (A = V ↔ x x A)
Distinct variable group:   x,A

Proof of Theorem eqv
StepHypRef Expression
1 dfcleq 2017 . 2 (A = V ↔ x(x Ax V))
2 vex 2537 . . . 4 x V
32tbt 236 . . 3 (x A ↔ (x Ax V))
43albii 1339 . 2 (x x Ax(x Ax V))
51, 4bitr4i 176 1 (A = V ↔ x x A)
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1226   = wceq 1228   wcel 1375  Vcvv 2534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-v 2536
This theorem is referenced by:  setindel  4203  dmi  4475
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