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Theorem sbcabel 2896
 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:   ,   ,
Allowed substitution hints:   (,)   ()   (,)   (,)

Proof of Theorem sbcabel
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2611 . 2
2 sbcexg 2869 . . . 4
3 sbcang 2858 . . . . . 6
4 sbcalg 2867 . . . . . . . . 9
5 sbcbig 2861 . . . . . . . . . . 11
6 sbcg 2884 . . . . . . . . . . . 12
76bibi1d 231 . . . . . . . . . . 11
85, 7bitrd 186 . . . . . . . . . 10
98albidv 1746 . . . . . . . . 9
104, 9bitrd 186 . . . . . . . 8
11 abeq2 2188 . . . . . . . . 9
1211sbcbii 2874 . . . . . . . 8
13 abeq2 2188 . . . . . . . 8
1410, 12, 133bitr4g 221 . . . . . . 7
15 sbcabel.1 . . . . . . . . 9
1615nfcri 2214 . . . . . . . 8
1716sbcgf 2882 . . . . . . 7
1814, 17anbi12d 457 . . . . . 6
193, 18bitrd 186 . . . . 5
2019exbidv 1747 . . . 4
212, 20bitrd 186 . . 3
22 df-clel 2078 . . . 4
2322sbcbii 2874 . . 3
24 df-clel 2078 . . 3
2521, 23, 243bitr4g 221 . 2
261, 25syl 14 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wb 103  wal 1283   wceq 1285  wex 1422   wcel 1434  cab 2068  wnfc 2207  cvv 2602  wsbc 2816 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817 This theorem is referenced by:  csbexga  3914
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