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Theorem sbidm 1807
Description: An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
Assertion
Ref Expression
sbidm  |-  ( [ y  /  x ] [ y  /  x ] ph  <->  [ y  /  x ] ph )

Proof of Theorem sbidm
StepHypRef Expression
1 df-sb 1721 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
21simplbi 272 . . . 4  |-  ( [ y  /  x ] ph  ->  ( x  =  y  ->  ph ) )
32sbimi 1722 . . 3  |-  ( [ y  /  x ] [ y  /  x ] ph  ->  [ y  /  x ] ( x  =  y  ->  ph )
)
4 sbequ8 1803 . . 3  |-  ( [ y  /  x ] ph 
<->  [ y  /  x ] ( x  =  y  ->  ph ) )
53, 4sylibr 133 . 2  |-  ( [ y  /  x ] [ y  /  x ] ph  ->  [ y  /  x ] ph )
6 ax-1 6 . . 3  |-  ( [ y  /  x ] ph  ->  ( x  =  y  ->  [ y  /  x ] ph )
)
7 sb1 1724 . . . 4  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
8 pm4.24 392 . . . . . . . 8  |-  ( E. x ( x  =  y  /\  ph )  <->  ( E. x ( x  =  y  /\  ph )  /\  E. x ( x  =  y  /\  ph ) ) )
9 ax-ie1 1454 . . . . . . . . 9  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x E. x
( x  =  y  /\  ph ) )
10919.41h 1648 . . . . . . . 8  |-  ( E. x ( ( x  =  y  /\  ph )  /\  E. x ( x  =  y  /\  ph ) )  <->  ( E. x ( x  =  y  /\  ph )  /\  E. x ( x  =  y  /\  ph ) ) )
118, 10bitr4i 186 . . . . . . 7  |-  ( E. x ( x  =  y  /\  ph )  <->  E. x ( ( x  =  y  /\  ph )  /\  E. x ( x  =  y  /\  ph ) ) )
12 ax-1 6 . . . . . . . . . 10  |-  ( ph  ->  ( x  =  y  ->  ph ) )
1312anim2i 339 . . . . . . . . 9  |-  ( ( x  =  y  /\  ph )  ->  ( x  =  y  /\  (
x  =  y  ->  ph ) ) )
1413anim1i 338 . . . . . . . 8  |-  ( ( ( x  =  y  /\  ph )  /\  E. x ( x  =  y  /\  ph )
)  ->  ( (
x  =  y  /\  ( x  =  y  ->  ph ) )  /\  E. x ( x  =  y  /\  ph )
) )
1514eximi 1564 . . . . . . 7  |-  ( E. x ( ( x  =  y  /\  ph )  /\  E. x ( x  =  y  /\  ph ) )  ->  E. x
( ( x  =  y  /\  ( x  =  y  ->  ph )
)  /\  E. x
( x  =  y  /\  ph ) ) )
1611, 15sylbi 120 . . . . . 6  |-  ( E. x ( x  =  y  /\  ph )  ->  E. x ( ( x  =  y  /\  ( x  =  y  ->  ph ) )  /\  E. x ( x  =  y  /\  ph )
) )
17 anass 398 . . . . . . 7  |-  ( ( ( x  =  y  /\  ( x  =  y  ->  ph ) )  /\  E. x ( x  =  y  /\  ph ) )  <->  ( x  =  y  /\  (
( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) ) )
1817exbii 1569 . . . . . 6  |-  ( E. x ( ( x  =  y  /\  (
x  =  y  ->  ph ) )  /\  E. x ( x  =  y  /\  ph )
)  <->  E. x ( x  =  y  /\  (
( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) ) )
1916, 18sylib 121 . . . . 5  |-  ( E. x ( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  (
( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) ) )
201anbi2i 452 . . . . . 6  |-  ( ( x  =  y  /\  [ y  /  x ] ph )  <->  ( x  =  y  /\  ( ( x  =  y  ->  ph )  /\  E. x
( x  =  y  /\  ph ) ) ) )
2120exbii 1569 . . . . 5  |-  ( E. x ( x  =  y  /\  [ y  /  x ] ph ) 
<->  E. x ( x  =  y  /\  (
( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) ) )
2219, 21sylibr 133 . . . 4  |-  ( E. x ( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  [
y  /  x ] ph ) )
237, 22syl 14 . . 3  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  [
y  /  x ] ph ) )
24 df-sb 1721 . . 3  |-  ( [ y  /  x ] [ y  /  x ] ph  <->  ( ( x  =  y  ->  [ y  /  x ] ph )  /\  E. x ( x  =  y  /\  [ y  /  x ] ph ) ) )
256, 23, 24sylanbrc 413 . 2  |-  ( [ y  /  x ] ph  ->  [ y  /  x ] [ y  /  x ] ph )
265, 25impbii 125 1  |-  ( [ y  /  x ] [ y  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   E.wex 1453   [wsb 1720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-sb 1721
This theorem is referenced by: (None)
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