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Theorem axdc3 8080
 Description: Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value . This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. (Contributed by Mario Carneiro, 27-Jan-2013.)
Hypothesis
Ref Expression
axdc3.1
Assertion
Ref Expression
axdc3
Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem axdc3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axdc3.1 . 2
2 feq1 5375 . . . . 5
3 fveq1 5524 . . . . . 6
43eqeq1d 2291 . . . . 5
5 fveq1 5524 . . . . . . . 8
6 fveq1 5524 . . . . . . . . 9
76fveq2d 5529 . . . . . . . 8
85, 7eleq12d 2351 . . . . . . 7
98ralbidv 2563 . . . . . 6
10 suceq 4457 . . . . . . . . 9
1110fveq2d 5529 . . . . . . . 8
12 fveq2 5525 . . . . . . . . 9
1312fveq2d 5529 . . . . . . . 8
1411, 13eleq12d 2351 . . . . . . 7
1514cbvralv 2764 . . . . . 6
169, 15syl6bb 252 . . . . 5
172, 4, 163anbi123d 1252 . . . 4
1817rexbidv 2564 . . 3
1918cbvabv 2402 . 2
20 eqid 2283 . 2
211, 19, 20axdc3lem4 8079 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934  wex 1528   wceq 1623   wcel 1684  cab 2269  wral 2543  wrex 2544  crab 2547  cvv 2788   cdif 3149  c0 3455  cpw 3625  csn 3640   cmpt 4077   csuc 4394  com 4656   cdm 4689   cres 4691  wf 5251  cfv 5255 This theorem is referenced by:  axdc4lem  8081 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-dc 8072 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479
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