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Theorem bnd2 3926
 Description: A variant of the Boundedness Axiom bnd 3925 that picks a subset out of a possibly proper class in which a property is true. (Contributed by NM, 4-Feb-2004.)
Hypothesis
Ref Expression
bnd2.1
Assertion
Ref Expression
bnd2
Distinct variable groups:   ,   ,,   ,,,
Allowed substitution hints:   (,)   ()

Proof of Theorem bnd2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rex 2312 . . . 4
21ralbii 2330 . . 3
3 bnd2.1 . . . 4
4 raleq 2505 . . . . 5
5 raleq 2505 . . . . . 6
65exbidv 1706 . . . . 5
74, 6imbi12d 223 . . . 4
8 bnd 3925 . . . 4
93, 7, 8vtocl 2608 . . 3
102, 9sylbi 114 . 2
11 vex 2560 . . . . 5
1211inex1 3891 . . . 4
13 inss2 3158 . . . . . . 7
14 sseq1 2966 . . . . . . 7
1513, 14mpbiri 157 . . . . . 6
1615biantrurd 289 . . . . 5
17 rexeq 2506 . . . . . . 7
18 elin 3126 . . . . . . . . . 10
1918anbi1i 431 . . . . . . . . 9
20 anass 381 . . . . . . . . 9
2119, 20bitri 173 . . . . . . . 8
2221rexbii2 2335 . . . . . . 7
2317, 22syl6bb 185 . . . . . 6
2423ralbidv 2326 . . . . 5
2516, 24bitr3d 179 . . . 4
2612, 25spcev 2647 . . 3
2726exlimiv 1489 . 2
2810, 27syl 14 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wceq 1243  wex 1381   wcel 1393  wral 2306  wrex 2307  cvv 2557   cin 2916   wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931 This theorem is referenced by: (None)
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