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Theorem reusv2 4607
 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:   ,,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem reusv2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfrab1 2957 . . . 4
2 nfcv 2612 . . . 4
3 nfv 1769 . . . 4
4 nfcsb1v 3365 . . . . 5
54nfel1 2626 . . . 4
6 csbeq1a 3358 . . . . 5
76eleq1d 2533 . . . 4
81, 2, 3, 5, 7cbvralf 2999 . . 3
9 rabid 2953 . . . . . 6
109imbi1i 332 . . . . 5
11 impexp 453 . . . . 5
1210, 11bitri 257 . . . 4
1312ralbii2 2821 . . 3
148, 13bitr3i 259 . 2
15 rabn0 3755 . 2
16 reusv2lem5 4606 . . 3
17 nfv 1769 . . . . . 6
184nfeq2 2627 . . . . . 6
196eqeq2d 2481 . . . . . 6
201, 2, 17, 18, 19cbvrexf 3000 . . . . 5
219anbi1i 709 . . . . . . 7
22 anass 661 . . . . . . 7
2321, 22bitri 257 . . . . . 6
2423rexbii2 2879 . . . . 5
2520, 24bitr3i 259 . . . 4
2625reubii 2963 . . 3
271, 2, 17, 18, 19cbvralf 2999 . . . . 5
289imbi1i 332 . . . . . . 7
29 impexp 453 . . . . . . 7
3028, 29bitri 257 . . . . . 6
3130ralbii2 2821 . . . . 5
3227, 31bitr3i 259 . . . 4
3332reubii 2963 . . 3
3416, 26, 333bitr3g 295 . 2
3514, 15, 34syl2anbr 488 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   wceq 1452   wcel 1904   wne 2641  wral 2756  wrex 2757  wreu 2758  crab 2760  csb 3349  c0 3722 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527  ax-pow 4579 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-nul 3723 This theorem is referenced by:  cdleme25dN  33994
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