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Theorem equsb3 2178
Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
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 2177 . . 3  |-  ( [ w  /  y ] y  =  z  <->  w  =  z )
21sbbii 1665 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  w ] w  =  z
)
3 nfv 1629 . . 3  |-  F/ w  y  =  z
43sbco2 2161 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  y ] y  =  z )
5 equsb3lem 2177 . 2  |-  ( [ x  /  w ]
w  =  z  <->  x  =  z )
62, 4, 53bitr3i 267 1  |-  ( [ x  /  y ] y  =  z  <->  x  =  z )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   [wsb 1658
This theorem is referenced by:  sb8eu  2299  sb8iota  5418  mo5f  23965  sbeqal1  27566
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-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
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