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Theorem aceq2 8000
 Description: Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
aceq2
Distinct variable group:   ,,,,,

Proof of Theorem aceq2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-ral 2710 . . . . 5
2 19.23v 1914 . . . . 5
31, 2bitri 241 . . . 4
4 biidd 229 . . . . 5
54cbvralv 2932 . . . 4
6 n0 3637 . . . . 5
7 eleq2 2497 . . . . . . . . 9
8 eleq2 2497 . . . . . . . . 9
97, 8anbi12d 692 . . . . . . . 8
109cbvrexv 2933 . . . . . . 7
1110reubii 2894 . . . . . 6
12 eleq1 2496 . . . . . . . . 9
1312anbi2d 685 . . . . . . . 8
1413rexbidv 2726 . . . . . . 7
1514cbvreuv 2934 . . . . . 6
1611, 15bitri 241 . . . . 5
176, 16imbi12i 317 . . . 4
183, 5, 173bitr4i 269 . . 3
1918ralbii 2729 . 2
2019exbii 1592 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wal 1549  wex 1550   wne 2599  wral 2705  wrex 2706  wreu 2707  c0 3628 This theorem is referenced by:  dfac7  8012  ac3  8342 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-v 2958  df-dif 3323  df-nul 3629
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