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Theorem sbidm 2162
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 2104 . . 3  |-  [ y  /  x ] y  =  x
2 sbequ12r 1946 . . . 4  |-  ( y  =  x  ->  ( [ y  /  x ] ph  <->  ph ) )
32sbimi 1665 . . 3  |-  ( [ y  /  x ]
y  =  x  ->  [ y  /  x ] ( [ y  /  x ] ph  <->  ph ) )
41, 3ax-mp 5 . 2  |-  [ y  /  x ] ( [ y  /  x ] ph  <->  ph )
5 sbbi 2143 . 2  |-  ( [ y  /  x ]
( [ y  /  x ] ph  <->  ph )  <->  ( [
y  /  x ] [ y  /  x ] ph  <->  [ y  /  x ] ph ) )
64, 5mpbi 201 1  |-  ( [ y  /  x ] [ y  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   [wsb 1659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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