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Theorem dfrex2fin 6673
Description: Relationship between universal and existential quantifiers over a finite set. Remark in Section 2.2.1 of [Pierik], p. 8. Although Pierik does not mention the decidability condition explicitly, it does say "only finitely many x to check" which means there must be some way of checking each value of x. (Contributed by Jim Kingdon, 11-Jul-2022.)
Assertion
Ref Expression
dfrex2fin  |-  ( ( A  e.  Fin  /\  A. x  e.  A DECID  ph )  ->  ( E. x  e.  A  ph  <->  -.  A. x  e.  A  -.  ph )
)
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem dfrex2fin
StepHypRef Expression
1 finexdc 6672 . 2  |-  ( ( A  e.  Fin  /\  A. x  e.  A DECID  ph )  -> DECID  E. x  e.  A  ph )
2 dfrex2dc 2372 . 2  |-  (DECID  E. x  e.  A  ph  ->  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph ) )
31, 2syl 14 1  |-  ( ( A  e.  Fin  /\  A. x  e.  A DECID  ph )  ->  ( E. x  e.  A  ph  <->  -.  A. x  e.  A  -.  ph )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104  DECID wdc 781    e. wcel 1439   A.wral 2360   E.wrex 2361   Fincfn 6511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-er 6306  df-en 6512  df-fin 6514
This theorem is referenced by: (None)
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