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Theorem sbcabel 3416
 Description: Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
Hypothesis
Ref Expression
sbcabel.1
Assertion
Ref Expression
sbcabel
Distinct variable groups:   ,   ,

Proof of Theorem sbcabel
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3118 . 2
2 sbcex2 3381 . . . 4
3 sbcan 3370 . . . . . 6
4 sbcal 3379 . . . . . . . . 9
5 sbcbig 3374 . . . . . . . . . . 11
6 sbcg 3401 . . . . . . . . . . . 12
76bibi1d 319 . . . . . . . . . . 11
85, 7bitrd 253 . . . . . . . . . 10
98albidv 1713 . . . . . . . . 9
104, 9syl5bb 257 . . . . . . . 8
11 abeq2 2581 . . . . . . . . 9
1211sbcbii 3387 . . . . . . . 8
13 abeq2 2581 . . . . . . . 8
1410, 12, 133bitr4g 288 . . . . . . 7
15 sbcabel.1 . . . . . . . . 9
1615nfcri 2612 . . . . . . . 8
1716sbcgf 3399 . . . . . . 7
1814, 17anbi12d 710 . . . . . 6
193, 18syl5bb 257 . . . . 5
2019exbidv 1714 . . . 4
212, 20syl5bb 257 . . 3
22 df-clel 2452 . . . 4
2322sbcbii 3387 . . 3
24 df-clel 2452 . . 3
2521, 23, 243bitr4g 288 . 2
261, 25syl 16 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  F/_wnfc 2605   cvv 3109  [.wsbc 3327 This theorem is referenced by:  csbexg  4584  csbexgOLD  4586 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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328
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