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Theorem rabsnifsb 4031
 Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 21-Jul-2019.)
Assertion
Ref Expression
rabsnifsb
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem rabsnifsb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elsni 3985 . . . . . . . 8
2 sbceq1a 3266 . . . . . . . . 9
32biimpd 212 . . . . . . . 8
41, 3syl 17 . . . . . . 7
54imdistani 704 . . . . . 6
65orcd 399 . . . . 5
72bicomd 206 . . . . . . . . 9
81, 7syl 17 . . . . . . . 8
98biimpd 212 . . . . . . 7
109imdistani 704 . . . . . 6
11 noel 3726 . . . . . . . 8
1211pm2.21i 136 . . . . . . 7
1312adantr 472 . . . . . 6
1410, 13jaoi 386 . . . . 5
156, 14impbii 192 . . . 4
1615abbii 2587 . . 3
17 nfv 1769 . . . 4
18 nfv 1769 . . . . . 6
19 nfsbc1v 3275 . . . . . 6
2018, 19nfan 2031 . . . . 5
21 nfv 1769 . . . . . 6
2219nfn 2003 . . . . . 6
2321, 22nfan 2031 . . . . 5
2420, 23nfor 2038 . . . 4
25 eleq1 2537 . . . . . 6
2625anbi1d 719 . . . . 5
27 eleq1 2537 . . . . . 6
2827anbi1d 719 . . . . 5
2926, 28orbi12d 724 . . . 4
3017, 24, 29cbvab 2594 . . 3
3116, 30eqtri 2493 . 2
32 df-rab 2765 . 2
33 df-if 3873 . 2
3431, 32, 333eqtr4i 2503 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wo 375   wa 376   wceq 1452   wcel 1904  cab 2457  crab 2760  wsbc 3255  c0 3722  cif 3872  csn 3959 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-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 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-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-nul 3723  df-if 3873  df-sn 3960 This theorem is referenced by:  rabsnif  4032  rabrsn  4033
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