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Theorem reusv2 4729
 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 2888 . . . 4
2 nfcv 2572 . . . 4
3 nfv 1629 . . . 4
4 nfcsb1v 3283 . . . . 5
54nfel1 2582 . . . 4
6 csbeq1a 3259 . . . . 5
76eleq1d 2502 . . . 4
81, 2, 3, 5, 7cbvralf 2926 . . 3
9 rabid 2884 . . . . . 6
109imbi1i 316 . . . . 5
11 impexp 434 . . . . 5
1210, 11bitri 241 . . . 4
1312ralbii2 2733 . . 3
148, 13bitr3i 243 . 2
15 rabn0 3647 . 2
16 reusv2lem5 4728 . . 3
17 nfv 1629 . . . . . 6
184nfeq2 2583 . . . . . 6
196eqeq2d 2447 . . . . . 6
201, 2, 17, 18, 19cbvrexf 2927 . . . . 5
219anbi1i 677 . . . . . . 7
22 anass 631 . . . . . . 7
2321, 22bitri 241 . . . . . 6
2423rexbii2 2734 . . . . 5
2520, 24bitr3i 243 . . . 4
2625reubii 2894 . . 3
271, 2, 17, 18, 19cbvralf 2926 . . . . 5
289imbi1i 316 . . . . . . 7
29 impexp 434 . . . . . . 7
3028, 29bitri 241 . . . . . 6
3130ralbii2 2733 . . . . 5
3227, 31bitr3i 243 . . . 4
3332reubii 2894 . . 3
3416, 26, 333bitr3g 279 . 2
3514, 15, 34syl2anbr 467 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725   wne 2599  wral 2705  wrex 2706  wreu 2707  crab 2709  csb 3251  c0 3628 This theorem is referenced by:  cdleme25dN  31153 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338  ax-pow 4377 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-nul 3629
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