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Theorem equsb3 2281
Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.) Remove dependency on ax-11 1937. (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 2280 . . 3  |-  ( [ w  /  y ] y  =  z  <->  w  =  z )
21sbbii 1812 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  w ] w  =  z
)
3 sbcom3 2260 . . 3  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  w ] [ x  /  y ] y  =  z )
4 nfv 1769 . . . 4  |-  F/ w [ x  /  y ] y  =  z
54sbf 2229 . . 3  |-  ( [ x  /  w ] [ x  /  y ] y  =  z  <->  [ x  /  y ] y  =  z )
63, 5bitri 257 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  y ] y  =  z )
7 equsb3lem 2280 . 2  |-  ( [ x  /  w ]
w  =  z  <->  x  =  z )
82, 6, 73bitr3i 283 1  |-  ( [ x  /  y ] y  =  z  <->  x  =  z )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   [wsb 1805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806
This theorem is referenced by:  sb8eu  2352  mo3  2356  sb8iota  5560  mo5f  28199  mptsnunlem  31810  wl-equsb3  31954  wl-mo3t  31975  wl-sb8eut  31976  frege55lem1b  36562  sbeqal1  36818
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