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Theorem subsym1 24866
Description: A symmetry with  [ x  /  y ].

See negsym1 24856 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 1313 . . . . . . . . . 10  |-  -.  F.
21intnan 880 . . . . . . . . 9  |-  -.  (
y  =  x  /\  F.  )
32nex 1542 . . . . . . . 8  |-  -.  E. y ( y  =  x  /\  F.  )
43intnan 880 . . . . . . 7  |-  -.  (
( y  =  x  ->  F.  )  /\  E. y ( y  =  x  /\  F.  )
)
5 df-sb 1630 . . . . . . 7  |-  ( [ x  /  y ]  F.  <->  ( ( y  =  x  ->  F.  )  /\  E. y ( y  =  x  /\  F.  ) ) )
64, 5mtbir 290 . . . . . 6  |-  -.  [
x  /  y ]  F.
76intnan 880 . . . . 5  |-  -.  (
y  =  x  /\  [ x  /  y ]  F.  )
87nex 1542 . . . 4  |-  -.  E. y ( y  =  x  /\  [ x  /  y ]  F.  )
98intnan 880 . . 3  |-  -.  (
( y  =  x  ->  [ x  / 
y ]  F.  )  /\  E. y ( y  =  x  /\  [
x  /  y ]  F.  ) )
10 df-sb 1630 . . 3  |-  ( [ x  /  y ] [ x  /  y ]  F.  <->  ( ( y  =  x  ->  [ x  /  y ]  F.  )  /\  E. y ( y  =  x  /\  [ x  /  y ]  F.  ) ) )
119, 10mtbir 290 . 2  |-  -.  [
x  /  y ] [ x  /  y ]  F.
1211pm2.21i 123 1  |-  ( [ x  /  y ] [ x  /  y ]  F.  ->  [ x  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    F. wfal 1308   E.wex 1528    = wceq 1623   [wsb 1629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-fal 1311  df-ex 1529  df-sb 1630
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