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Theorem ac5 8388
 Description: An Axiom of Choice equivalent: there exists a function (called a choice function) with domain that maps each nonempty member of the domain to an element of that member. Axiom AC of [BellMachover] p. 488. Note that the assertion that be a function is not necessary; see ac4 8386. (Contributed by NM, 29-Aug-1999.)
Hypothesis
Ref Expression
ac5.1
Assertion
Ref Expression
ac5
Distinct variable group:   ,,

Proof of Theorem ac5
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ac5.1 . 2
2 fneq2 5564 . . . 4
3 raleq 2910 . . . 4
42, 3anbi12d 693 . . 3
54exbidv 1637 . 2
6 dfac4 8034 . . 3 CHOICE
76axaci 8379 . 2
81, 5, 7vtocl 3012 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360  wex 1551   wceq 1653   wcel 1727   wne 2605  wral 2711  cvv 2962  c0 3613   wfn 5478  cfv 5483 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-ac2 8374 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ac 8028
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