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Theorem equsb3 2238
Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.) Remove dependency on ax-11 1896. (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 2237 . . 3  |-  ( [ w  /  y ] y  =  z  <->  w  =  z )
21sbbii 1797 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  w ] w  =  z
)
3 sbcom3 2216 . . 3  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  w ] [ x  /  y ] y  =  z )
4 nfv 1755 . . . 4  |-  F/ w [ x  /  y ] y  =  z
54sbf 2185 . . 3  |-  ( [ x  /  w ] [ x  /  y ] y  =  z  <->  [ x  /  y ] y  =  z )
63, 5bitri 252 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  y ] y  =  z )
7 equsb3lem 2237 . 2  |-  ( [ x  /  w ]
w  =  z  <->  x  =  z )
82, 6, 73bitr3i 278 1  |-  ( [ x  /  y ] y  =  z  <->  x  =  z )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   [wsb 1790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2063
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1658  df-nf 1662  df-sb 1791
This theorem is referenced by:  sb8eu  2309  mo3  2313  sb8iota  5515  mo5f  28062  mptsnunlem  31647  wl-equsb3  31791  wl-mo3t  31812  wl-sb8eut  31813  frege55lem1b  36404  sbeqal1  36661
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