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Theorem sbidm 2025
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 1975 . . 3  |-  [ y  /  x ] y  =  x
2 sbequ12r 1861 . . . 4  |-  ( y  =  x  ->  ( [ y  /  x ] ph  <->  ph ) )
32sbimi 1633 . . 3  |-  ( [ y  /  x ]
y  =  x  ->  [ y  /  x ] ( [ y  /  x ] ph  <->  ph ) )
41, 3ax-mp 8 . 2  |-  [ y  /  x ] ( [ y  /  x ] ph  <->  ph )
5 sbbi 2011 . 2  |-  ( [ y  /  x ]
( [ y  /  x ] ph  <->  ph )  <->  ( [
y  /  x ] [ y  /  x ] ph  <->  [ y  /  x ] ph ) )
64, 5mpbi 199 1  |-  ( [ y  /  x ] [ y  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [wsb 1629
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-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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