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Theorem ifnot 3720
Description: Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
Assertion
Ref Expression
ifnot  |-  if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A )

Proof of Theorem ifnot
StepHypRef Expression
1 notnot1 116 . . . 4  |-  ( ph  ->  -.  -.  ph )
2 iffalse 3689 . . . 4  |-  ( -. 
-.  ph  ->  if ( -.  ph ,  A ,  B )  =  B )
31, 2syl 16 . . 3  |-  ( ph  ->  if ( -.  ph ,  A ,  B )  =  B )
4 iftrue 3688 . . 3  |-  ( ph  ->  if ( ph ,  B ,  A )  =  B )
53, 4eqtr4d 2422 . 2  |-  ( ph  ->  if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A ) )
6 iftrue 3688 . . 3  |-  ( -. 
ph  ->  if ( -. 
ph ,  A ,  B )  =  A )
7 iffalse 3689 . . 3  |-  ( -. 
ph  ->  if ( ph ,  B ,  A )  =  A )
86, 7eqtr4d 2422 . 2  |-  ( -. 
ph  ->  if ( -. 
ph ,  A ,  B )  =  if ( ph ,  B ,  A ) )
95, 8pm2.61i 158 1  |-  if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649   ifcif 3682
This theorem is referenced by:  sadadd2lem2  12889  tmsxpsval2  18459  itg2uba  19502  lgsneg  20970  lgsdilem  20973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-if 3683
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