Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb2 Unicode version

Theorem sb2 2093
 Description: One direction of a simplified definition of substitution. The converse requires either a dv condition (sb6 2173) or a non-freeness hypothesis (sb6f 2126). (Contributed by NM, 13-May-1993.)
Assertion
Ref Expression
sb2

Proof of Theorem sb2
StepHypRef Expression
1 sp 1859 . 2
2 equs4 2035 . 2
3 df-sb 1740 . 2
41, 2, 3sylanbrc 664 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  E.wex 1612  [wsb 1739 This theorem is referenced by:  stdpc4  2094  sb3  2096  sb4b  2098  hbsb2  2099  hbsb2a  2101  hbsb2e  2102  equsb1  2107  equsb2  2108  dfsb2  2114  sbequi  2116  sb6f  2126  sbi1  2133  sb6  2173  iota4  5574  wl-lem-moexsb  30017  sbeqal1  31304 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-12 1854  ax-13 1999 This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-sb 1740
 Copyright terms: Public domain W3C validator