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Theorem zfac 8861
Description: Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 8860. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
zfac
Distinct variable group:   , , ,

Proof of Theorem zfac
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-ac 8860 . 2
2 equequ2 1799 . . . . . . . . . 10
32bibi2d 318 . . . . . . . . 9
4 elequ2 1823 . . . . . . . . . . . . 13
54anbi2d 703 . . . . . . . . . . . 12
6 elequ2 1823 . . . . . . . . . . . . 13
7 elequ1 1821 . . . . . . . . . . . . 13
86, 7anbi12d 710 . . . . . . . . . . . 12
95, 8anbi12d 710 . . . . . . . . . . 11
109cbvexv 2024 . . . . . . . . . 10
1110bibi1i 314 . . . . . . . . 9
123, 11syl6bb 261 . . . . . . . 8
1312albidv 1713 . . . . . . 7
14 elequ1 1821 . . . . . . . . . . . 12
1514anbi1d 704 . . . . . . . . . . 11
16 elequ1 1821 . . . . . . . . . . . 12
1716anbi1d 704 . . . . . . . . . . 11
1815, 17anbi12d 710 . . . . . . . . . 10
1918exbidv 1714 . . . . . . . . 9
20 equequ1 1798 . . . . . . . . 9
2119, 20bibi12d 321 . . . . . . . 8
2221cbvalv 2023 . . . . . . 7
2313, 22syl6bb 261 . . . . . 6
2423cbvexv 2024 . . . . 5
2524imbi2i 312 . . . 4
26252albii 1641 . . 3
2726exbii 1667 . 2
281, 27mpbi 208 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612
This theorem is referenced by:  axacndlem4  9009
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-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ac 8860
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617
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