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Theorem ifnot 3971
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 )
21iffalsed 3937 . . 3  |-  ( ph  ->  if ( -.  ph ,  A ,  B )  =  B )
3 iftrue 3932 . . 3  |-  ( ph  ->  if ( ph ,  B ,  A )  =  B )
42, 3eqtr4d 2487 . 2  |-  ( ph  ->  if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A ) )
5 iftrue 3932 . . 3  |-  ( -. 
ph  ->  if ( -. 
ph ,  A ,  B )  =  A )
6 iffalse 3935 . . 3  |-  ( -. 
ph  ->  if ( ph ,  B ,  A )  =  A )
75, 6eqtr4d 2487 . 2  |-  ( -. 
ph  ->  if ( -. 
ph ,  A ,  B )  =  if ( ph ,  B ,  A ) )
84, 7pm2.61i 164 1  |-  if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1383   ifcif 3926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-if 3927
This theorem is referenced by:  suppsnop  6917  sadadd2lem2  14082  maducoeval2  19120  tmsxpsval2  21020  itg2uba  22128  lgsneg  23572  lgsdilem  23575  sgnneg  28457  itgaddnclem2  30050  ftc1anclem5  30070  bj-xpimasn  34395
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