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Theorem sbidm 2160
Description: An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
sbidm  |-  ( [ y  /  x ] [ y  /  x ] ph  <->  [ y  /  x ] ph )

Proof of Theorem sbidm
StepHypRef Expression
1 equsb2 2114 . . 3  |-  [ y  /  x ] y  =  x
2 sbequ12r 1945 . . . 4  |-  ( y  =  x  ->  ( [ y  /  x ] ph  <->  ph ) )
32sbimi 1664 . . 3  |-  ( [ y  /  x ]
y  =  x  ->  [ y  /  x ] ( [ y  /  x ] ph  <->  ph ) )
41, 3ax-mp 8 . 2  |-  [ y  /  x ] ( [ y  /  x ] ph  <->  ph )
5 sbbi 2145 . 2  |-  ( [ y  /  x ]
( [ y  /  x ] ph  <->  ph )  <->  ( [
y  /  x ] [ y  /  x ] ph  <->  [ y  /  x ] ph ) )
64, 5mpbi 200 1  |-  ( [ y  /  x ] [ y  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   [wsb 1658
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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