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Theorem equsb3 2162
Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.) Remove dependency on ax-11 1828. (Revised by Wolf Lammen, 21-Sep-2018.)
Assertion
Ref Expression
equsb3  |-  ( [ x  /  y ] y  =  z  <->  x  =  z )
Distinct variable group:    y, z

Proof of Theorem equsb3
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 equsb3lem 2161 . . 3  |-  ( [ w  /  y ] y  =  z  <->  w  =  z )
21sbbii 1733 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  w ] w  =  z
)
3 sbcom3 2139 . . 3  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  w ] [ x  /  y ] y  =  z )
4 nfv 1694 . . . 4  |-  F/ w [ x  /  y ] y  =  z
54sbf 2107 . . 3  |-  ( [ x  /  w ] [ x  /  y ] y  =  z  <->  [ x  /  y ] y  =  z )
63, 5bitri 249 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  y ] y  =  z )
7 equsb3lem 2161 . 2  |-  ( [ x  /  w ]
w  =  z  <->  x  =  z )
82, 6, 73bitr3i 275 1  |-  ( [ x  /  y ] y  =  z  <->  x  =  z )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   [wsb 1726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-12 1840  ax-13 1985
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1600  df-nf 1604  df-sb 1727
This theorem is referenced by:  sb8eu  2304  sb8euOLD  2305  mo3  2309  sb8iota  5548  mo5f  27255  wl-equsb3  29979  wl-mo3t  29996  wl-sb8eut  29997  sbeqal1  31258  frege55lem1b  37590
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