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Theorem subsym1 29819
Description: A symmetry with  [ x  /  y ].

See negsym1 29809 for more information. (Contributed by Anthony Hart, 11-Sep-2011.)

Assertion
Ref Expression
subsym1  |-  ( [ x  /  y ] [ x  /  y ] F.  ->  [ x  /  y ] ph )

Proof of Theorem subsym1
StepHypRef Expression
1 fal 1386 . . . . . . . . . 10  |-  -. F.
21intnan 912 . . . . . . . . 9  |-  -.  (
y  =  x  /\ F.  )
32nex 1610 . . . . . . . 8  |-  -.  E. y ( y  =  x  /\ F.  )
43intnan 912 . . . . . . 7  |-  -.  (
( y  =  x  -> F.  )  /\  E. y ( y  =  x  /\ F.  )
)
5 df-sb 1712 . . . . . . 7  |-  ( [ x  /  y ] F.  <->  ( ( y  =  x  -> F.  )  /\  E. y ( y  =  x  /\ F.  ) ) )
64, 5mtbir 299 . . . . . 6  |-  -.  [
x  /  y ] F.
76intnan 912 . . . . 5  |-  -.  (
y  =  x  /\  [ x  /  y ] F.  )
87nex 1610 . . . 4  |-  -.  E. y ( y  =  x  /\  [ x  /  y ] F.  )
98intnan 912 . . 3  |-  -.  (
( y  =  x  ->  [ x  / 
y ] F.  )  /\  E. y ( y  =  x  /\  [
x  /  y ] F.  ) )
10 df-sb 1712 . . 3  |-  ( [ x  /  y ] [ x  /  y ] F.  <->  ( ( y  =  x  ->  [ x  /  y ] F.  )  /\  E. y ( y  =  x  /\  [ x  /  y ] F.  ) ) )
119, 10mtbir 299 . 2  |-  -.  [
x  /  y ] [ x  /  y ] F.
1211pm2.21i 131 1  |-  ( [ x  /  y ] [ x  /  y ] F.  ->  [ x  /  y ] ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   F. wfal 1384   E.wex 1596   [wsb 1711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-fal 1385  df-ex 1597  df-sb 1712
This theorem is referenced by: (None)
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