Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcabel Structured version   Unicode version

Theorem sbcabel 3354
 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 3067 . 2
2 sbcex2 3327 . . . 4
3 sbcan 3319 . . . . . 6
4 sbcal 3326 . . . . . . . . 9
5 sbcbig 3321 . . . . . . . . . . 11
6 sbcg 3342 . . . . . . . . . . . 12
76bibi1d 317 . . . . . . . . . . 11
85, 7bitrd 253 . . . . . . . . . 10
98albidv 1734 . . . . . . . . 9
104, 9syl5bb 257 . . . . . . . 8
11 abeq2 2526 . . . . . . . . 9
1211sbcbii 3332 . . . . . . . 8
13 abeq2 2526 . . . . . . . 8
1410, 12, 133bitr4g 288 . . . . . . 7
15 sbcabel.1 . . . . . . . . 9
1615nfcri 2557 . . . . . . . 8
1716sbcgf 3340 . . . . . . 7
1814, 17anbi12d 709 . . . . . 6
193, 18syl5bb 257 . . . . 5
2019exbidv 1735 . . . 4
212, 20syl5bb 257 . . 3
22 df-clel 2397 . . . 4
2322sbcbii 3332 . . 3
24 df-clel 2397 . . 3
2521, 23, 243bitr4g 288 . 2
261, 25syl 17 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 367  wal 1403   wceq 1405  wex 1633   wcel 1842  cab 2387  wnfc 2550  cvv 3058  wsbc 3276 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-sbc 3277 This theorem is referenced by:  csbexg  4527
 Copyright terms: Public domain W3C validator