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Theorem ifnot 3943
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 3908 . . . 4  |-  ( -. 
-.  ph  ->  if ( -.  ph ,  A ,  B )  =  B )
31, 2syl 16 . . 3  |-  ( ph  ->  if ( -.  ph ,  A ,  B )  =  B )
4 iftrue 3906 . . 3  |-  ( ph  ->  if ( ph ,  B ,  A )  =  B )
53, 4eqtr4d 2498 . 2  |-  ( ph  ->  if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A ) )
6 iftrue 3906 . . 3  |-  ( -. 
ph  ->  if ( -. 
ph ,  A ,  B )  =  A )
7 iffalse 3908 . . 3  |-  ( -. 
ph  ->  if ( ph ,  B ,  A )  =  A )
86, 7eqtr4d 2498 . 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 1370   ifcif 3900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-if 3901
This theorem is referenced by:  suppsnop  6815  sadadd2lem2  13765  maducoeval2  18579  tmsxpsval2  20247  itg2uba  21355  lgsneg  22792  lgsdilem  22795  sgnneg  27068  itgaddnclem2  28600  ftc1anclem5  28620  bj-xpimasn  32780
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