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Theorem sbcabel 3102
 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 2830 . 2
2 sbcexg 3075 . . . 4
3 sbcang 3068 . . . . . 6
4 abeq2 2421 . . . . . . . . . 10
54sbcbii 3080 . . . . . . . . 9
6 sbcalg 3073 . . . . . . . . . 10
7 sbcbig 3071 . . . . . . . . . . . 12
8 sbcg 3090 . . . . . . . . . . . . 13
98bibi1d 310 . . . . . . . . . . . 12
107, 9bitrd 244 . . . . . . . . . . 11
1110albidv 1616 . . . . . . . . . 10
126, 11bitrd 244 . . . . . . . . 9
135, 12syl5bb 248 . . . . . . . 8
14 abeq2 2421 . . . . . . . 8
1513, 14syl6bbr 254 . . . . . . 7
16 sbcabel.1 . . . . . . . . 9
1716nfcri 2446 . . . . . . . 8
1817sbcgf 3088 . . . . . . 7
1915, 18anbi12d 691 . . . . . 6
203, 19bitrd 244 . . . . 5
2120exbidv 1617 . . . 4
222, 21bitrd 244 . . 3
23 df-clel 2312 . . . 4
2423sbcbii 3080 . . 3
25 df-clel 2312 . . 3
2622, 24, 253bitr4g 279 . 2
271, 26syl 15 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358  wal 1531  wex 1532   wceq 1633   wcel 1701  cab 2302  wnfc 2439  cvv 2822  wsbc 3025 This theorem is referenced by:  csbexg  3125 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-sbc 3026
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