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

Theorem sbequi 2116
 Description: An equality theorem for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 15-Sep-2018.)
Assertion
Ref Expression
sbequi

Proof of Theorem sbequi
StepHypRef Expression
1 equtr 1796 . . 3
2 sbequ2 1741 . . . 4
3 sbequ1 1991 . . . 4
42, 3syl9 71 . . 3
51, 4syld 44 . 2
6 ax13 2047 . . 3
7 sp 1859 . . . . . 6
87con3i 135 . . . . 5
9 sb4 2097 . . . . 5
108, 9syl 16 . . . 4
11 equequ2 1799 . . . . . . . 8
1211biimprd 223 . . . . . . 7
1312imim1d 75 . . . . . 6
1413al2imi 1636 . . . . 5
15 sb2 2093 . . . . 5
1614, 15syl6 33 . . . 4
1710, 16syl9 71 . . 3
186, 17syld 44 . 2
195, 18pm2.61i 164 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  A.wal 1393  [wsb 1739 This theorem is referenced by:  sbequ  2117 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-an 371  df-ex 1613  df-nf 1617  df-sb 1740
 Copyright terms: Public domain W3C validator