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Theorem aceq2 8521
 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 2812 . . . . 5
2 19.23v 1760 . . . . 5
31, 2bitri 249 . . . 4
4 biidd 237 . . . . 5
54cbvralv 3084 . . . 4
6 n0 3794 . . . . 5
7 eleq2 2530 . . . . . . . . 9
8 eleq2 2530 . . . . . . . . 9
97, 8anbi12d 710 . . . . . . . 8
109cbvrexv 3085 . . . . . . 7
1110reubii 3044 . . . . . 6
12 eleq1 2529 . . . . . . . . 9
1312anbi2d 703 . . . . . . . 8
1413rexbidv 2968 . . . . . . 7
1514cbvreuv 3086 . . . . . 6
1611, 15bitri 249 . . . . 5
176, 16imbi12i 326 . . . 4
183, 5, 173bitr4i 277 . . 3
1918ralbii 2888 . 2
2019exbii 1667 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  =/=wne 2652  A.wral 2807  E.wrex 2808  E!wreu 2809   c0 3784 This theorem is referenced by:  dfac7  8533  ac3  8863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-v 3111  df-dif 3478  df-nul 3785
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