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Theorem reusv2 4658
Description: Two ways to express single-valuedness of a class expression ( ) that is constant for those such that . The first antecedent ensures that the constant value belongs to the existential uniqueness domain , and the second ensures that ( ) is evaluated for at least one . (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reusv2
Distinct variable groups:   , ,   ,   ,   ,

Proof of Theorem reusv2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfrab1 3038 . . . 4
2 nfcv 2619 . . . 4
3 nfv 1707 . . . 4
4 nfcsb1v 3450 . . . . 5
54nfel1 2635 . . . 4
6 csbeq1a 3443 . . . . 5
76eleq1d 2526 . . . 4
81, 2, 3, 5, 7cbvralf 3078 . . 3
9 rabid 3034 . . . . . 6
109imbi1i 325 . . . . 5
11 impexp 446 . . . . 5
1210, 11bitri 249 . . . 4
1312ralbii2 2886 . . 3
148, 13bitr3i 251 . 2
15 rabn0 3805 . 2
16 reusv2lem5 4657 . . 3
17 nfv 1707 . . . . . 6
184nfeq2 2636 . . . . . 6
196eqeq2d 2471 . . . . . 6
201, 2, 17, 18, 19cbvrexf 3079 . . . . 5
219anbi1i 695 . . . . . . 7
22 anass 649 . . . . . . 7
2321, 22bitri 249 . . . . . 6
2423rexbii2 2957 . . . . 5
2520, 24bitr3i 251 . . . 4
2625reubii 3044 . . 3
271, 2, 17, 18, 19cbvralf 3078 . . . . 5
289imbi1i 325 . . . . . . 7
29 impexp 446 . . . . . . 7
3028, 29bitri 249 . . . . . 6
3130ralbii2 2886 . . . . 5
3227, 31bitr3i 251 . . . 4
3332reubii 3044 . . 3
3416, 26, 333bitr3g 287 . 2
3514, 15, 34syl2anbr 480 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808  E!wreu 2809  {crab 2811  [_csb 3434   c0 3784
This theorem is referenced by:  cdleme25dN  36082
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-ext 2435  ax-nul 4581  ax-pow 4630
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  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 3435  df-dif 3478  df-nul 3785
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