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Theorem ac5 8874
 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 8872. (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 5676 . . . 4
3 raleq 3054 . . . 4
42, 3anbi12d 710 . . 3
54exbidv 1715 . 2
6 dfac4 8520 . . 3 CHOICE
76axaci 8865 . 2
81, 5, 7vtocl 3161 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1395  wex 1613   wcel 1819   wne 2652  wral 2807  cvv 3109  c0 3793   wfn 5589  cfv 5594 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-ac2 8860 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  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-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ac 8514 This theorem is referenced by: (None)
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