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Theorem sbcabel 3239
 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 2965 . 2
2 sbcexg 3212 . . . 4
3 sbcang 3205 . . . . . 6
4 sbcalg 3210 . . . . . . . . 9
5 sbcbig 3208 . . . . . . . . . . 11
6 sbcg 3227 . . . . . . . . . . . 12
76bibi1d 312 . . . . . . . . . . 11
85, 7bitrd 246 . . . . . . . . . 10
98albidv 1636 . . . . . . . . 9
104, 9bitrd 246 . . . . . . . 8
11 abeq2 2542 . . . . . . . . 9
1211sbcbii 3217 . . . . . . . 8
13 abeq2 2542 . . . . . . . 8
1410, 12, 133bitr4g 281 . . . . . . 7
15 sbcabel.1 . . . . . . . . 9
1615nfcri 2567 . . . . . . . 8
1716sbcgf 3225 . . . . . . 7
1814, 17anbi12d 693 . . . . . 6
193, 18bitrd 246 . . . . 5
2019exbidv 1637 . . . 4
212, 20bitrd 246 . . 3
22 df-clel 2433 . . . 4
2322sbcbii 3217 . . 3
24 df-clel 2433 . . 3
2521, 23, 243bitr4g 281 . 2
261, 25syl 16 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wal 1550  wex 1551   wceq 1653   wcel 1726  cab 2423  wnfc 2560  cvv 2957  wsbc 3162 This theorem is referenced by:  csbexg  3262 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-sbc 3163
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