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

Theorem sbn 2132
Description: Negation inside and outside of substitution are equivalent. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)
Assertion
Ref Expression
sbn

Proof of Theorem sbn
StepHypRef Expression
1 df-sb 1740 . . 3
2 exanali 1670 . . . 4
32anbi2i 694 . . 3
4 annim 425 . . 3
51, 3, 43bitri 271 . 2
6 dfsb3 2115 . 2
75, 6xchbinxr 311 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  [wsb 1739
This theorem is referenced by:  sbi2  2134  sbor  2139  sban  2140  sbex  2207  sbcng  3368  difab  3766  wl-sb8et  30001  pm13.196a  31321  bj-abfal  34474
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-12 1854  ax-13 1999
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1613  df-nf 1617  df-sb 1740
  Copyright terms: Public domain W3C validator