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Theorem sbcabel 3123
Description: Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
Hypothesis
Ref Expression
sbcabel.1  F/_
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 2867 . 2
2 sbcexg 3096 . . . 4  [.  ].  [.  ].
3 sbcang 3089 . . . . . 6  [.  ].  [.  ].  [.  ].
4 abeq2 2458 . . . . . . . . . 10
54sbcbii 3101 . . . . . . . . 9  [.  ]. 
[.  ].
6 sbcalg 3094 . . . . . . . . . 10  [.  ].  [.  ].
7 sbcbig 3092 . . . . . . . . . . . 12  [.  ].  [.  ].  [.  ].
8 sbcg 3111 . . . . . . . . . . . . 13  [.  ].
98bibi1d 310 . . . . . . . . . . . 12  [.  ].  [.  ].  [.  ].
107, 9bitrd 244 . . . . . . . . . . 11  [.  ].  [.  ].
1110albidv 1625 . . . . . . . . . 10  [.  ].  [.  ].
126, 11bitrd 244 . . . . . . . . 9  [.  ].  [.  ].
135, 12syl5bb 248 . . . . . . . 8  [.  ].  [.  ].
14 abeq2 2458 . . . . . . . 8 
[.  ].  [.  ].
1513, 14syl6bbr 254 . . . . . . 7  [.  ].  [.  ].
16 sbcabel.1 . . . . . . . . 9  F/_
1716nfcri 2483 . . . . . . . 8  F/
1817sbcgf 3109 . . . . . . 7  [.  ].
1915, 18anbi12d 691 . . . . . 6  [.  ].  [.  ]. 
[.  ].
203, 19bitrd 244 . . . . 5  [.  ].  [.  ].
2120exbidv 1626 . . . 4  [.  ]. 
[.  ].
222, 21bitrd 244 . . 3  [.  ]. 
[.  ].
23 df-clel 2349 . . . 4
2423sbcbii 3101 . . 3  [.  ]. 
[.  ].
25 df-clel 2349 . . 3  [.  ]. 
[.  ].
2622, 24, 253bitr4g 279 . 2  [.  ]. 
[.  ].
271, 26syl 15 1  [.  ]. 
[.  ].
Colors of variables: wff setvar class
Syntax hints:   wi 4   wb 176   wa 358  wal 1540  wex 1541   wceq 1642   wcel 1710  cab 2339   F/_wnfc 2476  cvv 2859   [.wsbc 3046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047
This theorem is referenced by:  csbexg  3146
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