Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  elsb3 Structured version   Unicode version

Theorem elsb3 2178
 Description: Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
elsb3
Distinct variable group:   ,

Proof of Theorem elsb3
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1629 . . 3
21sbco2 2161 . 2
3 nfv 1629 . . . 4
4 elequ1 1728 . . . 4
53, 4sbie 2122 . . 3
65sbbii 1665 . 2
7 nfv 1629 . . 3
8 elequ1 1728 . . 3
97, 8sbie 2122 . 2
102, 6, 93bitr3i 267 1
 Colors of variables: wff set class Syntax hints:   wb 177  wsb 1658 This theorem is referenced by:  cvjust  2430 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
 Copyright terms: Public domain W3C validator