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Theorem aceq2 7762
 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 2561 . . . . 5
2 19.23v 1844 . . . . 5
31, 2bitri 240 . . . 4
4 biidd 228 . . . . 5
54cbvralv 2777 . . . 4
6 n0 3477 . . . . 5
7 eleq2 2357 . . . . . . . . 9
8 eleq2 2357 . . . . . . . . 9
97, 8anbi12d 691 . . . . . . . 8
109cbvrexv 2778 . . . . . . 7
1110reubii 2739 . . . . . 6
12 eleq1 2356 . . . . . . . . 9
1312anbi2d 684 . . . . . . . 8
1413rexbidv 2577 . . . . . . 7
1514cbvreuv 2779 . . . . . 6
1611, 15bitri 240 . . . . 5
176, 16imbi12i 316 . . . 4
183, 5, 173bitr4i 268 . . 3
1918ralbii 2580 . 2
2019exbii 1572 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358  wal 1530  wex 1531   wceq 1632   wcel 1696   wne 2459  wral 2556  wrex 2557  wreu 2558  c0 3468 This theorem is referenced by:  dfac7  7774  ac3  8104 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-v 2803  df-dif 3168  df-nul 3469
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