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

Theorem sbcralt 3408
Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
Assertion
Ref Expression
sbcralt
Distinct variable groups:   ,   ,

Proof of Theorem sbcralt
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbcco 3350 . 2
2 simpl 457 . . 3
3 sbsbc 3331 . . . . 5
4 nfcv 2619 . . . . . . 7
5 nfs1v 2181 . . . . . . 7
64, 5nfral 2843 . . . . . 6
7 sbequ12 1992 . . . . . . 7
87ralbidv 2896 . . . . . 6
96, 8sbie 2149 . . . . 5
103, 9bitr3i 251 . . . 4
11 nfnfc1 2622 . . . . . . 7
12 nfcvd 2620 . . . . . . . 8
13 id 22 . . . . . . . 8
1412, 13nfeqd 2626 . . . . . . 7
1511, 14nfan1 1927 . . . . . 6
16 dfsbcq2 3330 . . . . . . 7
1716adantl 466 . . . . . 6
1815, 17ralbid 2891 . . . . 5
1918adantll 713 . . . 4
2010, 19syl5bb 257 . . 3
212, 20sbcied 3364 . 2
221, 21syl5bbr 259 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  [wsb 1739  e.wcel 1818  F/_wnfc 2605  A.wral 2807  [.wsbc 3327
This theorem is referenced by:  sbcrextOLD  3409  sbcrext  3410  sbcralg  3411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-sbc 3328
  Copyright terms: Public domain W3C validator