ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbidm Unicode version

Theorem sbidm 1774
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 1688 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
21simplbi 268 . . . 4  |-  ( [ y  /  x ] ph  ->  ( x  =  y  ->  ph ) )
32sbimi 1689 . . 3  |-  ( [ y  /  x ] [ y  /  x ] ph  ->  [ y  /  x ] ( x  =  y  ->  ph )
)
4 sbequ8 1770 . . 3  |-  ( [ y  /  x ] ph 
<->  [ y  /  x ] ( x  =  y  ->  ph ) )
53, 4sylibr 132 . 2  |-  ( [ y  /  x ] [ y  /  x ] ph  ->  [ y  /  x ] ph )
6 ax-1 5 . . 3  |-  ( [ y  /  x ] ph  ->  ( x  =  y  ->  [ y  /  x ] ph )
)
7 sb1 1691 . . . 4  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
8 pm4.24 387 . . . . . . . 8  |-  ( E. x ( x  =  y  /\  ph )  <->  ( E. x ( x  =  y  /\  ph )  /\  E. x ( x  =  y  /\  ph ) ) )
9 ax-ie1 1423 . . . . . . . . 9  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x E. x
( x  =  y  /\  ph ) )
10919.41h 1616 . . . . . . . 8  |-  ( E. x ( ( x  =  y  /\  ph )  /\  E. x ( x  =  y  /\  ph ) )  <->  ( E. x ( x  =  y  /\  ph )  /\  E. x ( x  =  y  /\  ph ) ) )
118, 10bitr4i 185 . . . . . . 7  |-  ( E. x ( x  =  y  /\  ph )  <->  E. x ( ( x  =  y  /\  ph )  /\  E. x ( x  =  y  /\  ph ) ) )
12 ax-1 5 . . . . . . . . . 10  |-  ( ph  ->  ( x  =  y  ->  ph ) )
1312anim2i 334 . . . . . . . . 9  |-  ( ( x  =  y  /\  ph )  ->  ( x  =  y  /\  (
x  =  y  ->  ph ) ) )
1413anim1i 333 . . . . . . . 8  |-  ( ( ( x  =  y  /\  ph )  /\  E. x ( x  =  y  /\  ph )
)  ->  ( (
x  =  y  /\  ( x  =  y  ->  ph ) )  /\  E. x ( x  =  y  /\  ph )
) )
1514eximi 1532 . . . . . . 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 119 . . . . . 6  |-  ( E. x ( x  =  y  /\  ph )  ->  E. x ( ( x  =  y  /\  ( x  =  y  ->  ph ) )  /\  E. x ( x  =  y  /\  ph )
) )
17 anass 393 . . . . . . 7  |-  ( ( ( x  =  y  /\  ( x  =  y  ->  ph ) )  /\  E. x ( x  =  y  /\  ph ) )  <->  ( x  =  y  /\  (
( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) ) )
1817exbii 1537 . . . . . 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 120 . . . . 5  |-  ( E. x ( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  (
( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) ) )
201anbi2i 445 . . . . . 6  |-  ( ( x  =  y  /\  [ y  /  x ] ph )  <->  ( x  =  y  /\  ( ( x  =  y  ->  ph )  /\  E. x
( x  =  y  /\  ph ) ) ) )
2120exbii 1537 . . . . 5  |-  ( E. x ( x  =  y  /\  [ y  /  x ] ph ) 
<->  E. x ( x  =  y  /\  (
( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) ) )
2219, 21sylibr 132 . . . 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 1688 . . 3  |-  ( [ y  /  x ] [ y  /  x ] ph  <->  ( ( x  =  y  ->  [ y  /  x ] ph )  /\  E. x ( x  =  y  /\  [ y  /  x ] ph ) ) )
256, 23, 24sylanbrc 408 . 2  |-  ( [ y  /  x ] ph  ->  [ y  /  x ] [ y  /  x ] ph )
265, 25impbii 124 1  |-  ( [ y  /  x ] [ y  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   E.wex 1422   [wsb 1687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1688
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator