MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elsb4 Structured version   Visualization version   Unicode version

Theorem elsb4 2274
Description: Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
elsb4  |-  ( [ x  /  y ] z  e.  y  <->  z  e.  x )
Distinct variable group:    y, z

Proof of Theorem elsb4
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1771 . . 3  |-  F/ y  z  e.  w
21sbco2 2254 . 2  |-  ( [ x  /  y ] [ y  /  w ] z  e.  w  <->  [ x  /  w ]
z  e.  w )
3 nfv 1771 . . . 4  |-  F/ w  z  e.  y
4 elequ2 1911 . . . 4  |-  ( w  =  y  ->  (
z  e.  w  <->  z  e.  y ) )
53, 4sbie 2247 . . 3  |-  ( [ y  /  w ]
z  e.  w  <->  z  e.  y )
65sbbii 1814 . 2  |-  ( [ x  /  y ] [ y  /  w ] z  e.  w  <->  [ x  /  y ] z  e.  y )
7 nfv 1771 . . 3  |-  F/ w  z  e.  x
8 elequ2 1911 . . 3  |-  ( w  =  x  ->  (
z  e.  w  <->  z  e.  x ) )
97, 8sbie 2247 . 2  |-  ( [ x  /  w ]
z  e.  w  <->  z  e.  x )
102, 6, 93bitr3i 283 1  |-  ( [ x  /  y ] z  e.  y  <->  z  e.  x )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   [wsb 1807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808
This theorem is referenced by:  nfnid  4642
  Copyright terms: Public domain W3C validator