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Theorem ifnot 3616
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 114 . . . 4  |-  ( ph  ->  -.  -.  ph )
2 iffalse 3585 . . . 4  |-  ( -. 
-.  ph  ->  if ( -.  ph ,  A ,  B )  =  B )
31, 2syl 15 . . 3  |-  ( ph  ->  if ( -.  ph ,  A ,  B )  =  B )
4 iftrue 3584 . . 3  |-  ( ph  ->  if ( ph ,  B ,  A )  =  B )
53, 4eqtr4d 2331 . 2  |-  ( ph  ->  if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A ) )
6 iftrue 3584 . . 3  |-  ( -. 
ph  ->  if ( -. 
ph ,  A ,  B )  =  A )
7 iffalse 3585 . . 3  |-  ( -. 
ph  ->  if ( ph ,  B ,  A )  =  A )
86, 7eqtr4d 2331 . 2  |-  ( -. 
ph  ->  if ( -. 
ph ,  A ,  B )  =  if ( ph ,  B ,  A ) )
95, 8pm2.61i 156 1  |-  if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632   ifcif 3578
This theorem is referenced by:  sadadd2lem2  12657  tmsxpsval2  18101  itg2uba  19114  lgsneg  20574  lgsdilem  20577
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-if 3579
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