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Theorem equsb3 2041
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 2040 . . 3  |-  ( [ w  /  y ] y  =  z  <->  w  =  z )
21sbbii 1634 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  w ] w  =  z
)
3 nfv 1605 . . 3  |-  F/ w  y  =  z
43sbco2 2026 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  z  <->  [ x  /  y ] y  =  z )
5 equsb3lem 2040 . 2  |-  ( [ x  /  w ]
w  =  z  <->  x  =  z )
62, 4, 53bitr3i 266 1  |-  ( [ x  /  y ] y  =  z  <->  x  =  z )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [wsb 1629
This theorem is referenced by:  sb8eu  2161  sb8iota  5226  mo5f  23143  sbeqal1  27597
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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