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Theorem ifnot 3990
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 122 . . . 4  |-  ( ph  ->  -.  -.  ph )
2 iffalse 3954 . . . 4  |-  ( -. 
-.  ph  ->  if ( -.  ph ,  A ,  B )  =  B )
31, 2syl 16 . . 3  |-  ( ph  ->  if ( -.  ph ,  A ,  B )  =  B )
4 iftrue 3951 . . 3  |-  ( ph  ->  if ( ph ,  B ,  A )  =  B )
53, 4eqtr4d 2511 . 2  |-  ( ph  ->  if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A ) )
6 iftrue 3951 . . 3  |-  ( -. 
ph  ->  if ( -. 
ph ,  A ,  B )  =  A )
7 iffalse 3954 . . 3  |-  ( -. 
ph  ->  if ( ph ,  B ,  A )  =  A )
86, 7eqtr4d 2511 . 2  |-  ( -. 
ph  ->  if ( -. 
ph ,  A ,  B )  =  if ( ph ,  B ,  A ) )
95, 8pm2.61i 164 1  |-  if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379   ifcif 3945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-if 3946
This theorem is referenced by:  suppsnop  6927  sadadd2lem2  13976  maducoeval2  19011  tmsxpsval2  20910  itg2uba  22018  lgsneg  23460  lgsdilem  23463  sgnneg  28295  itgaddnclem2  29992  ftc1anclem5  30012  bj-xpimasn  33949
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