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Theorem ifnotdc 3571
Description: Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
Assertion
Ref Expression
ifnotdc  |-  (DECID  ph  ->  if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A )
)

Proof of Theorem ifnotdc
StepHypRef Expression
1 df-dc 835 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 notnot 629 . . . . 5  |-  ( ph  ->  -.  -.  ph )
32iffalsed 3544 . . . 4  |-  ( ph  ->  if ( -.  ph ,  A ,  B )  =  B )
4 iftrue 3539 . . . 4  |-  ( ph  ->  if ( ph ,  B ,  A )  =  B )
53, 4eqtr4d 2213 . . 3  |-  ( ph  ->  if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A ) )
6 iftrue 3539 . . . 4  |-  ( -. 
ph  ->  if ( -. 
ph ,  A ,  B )  =  A )
7 iffalse 3542 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  B ,  A )  =  A )
86, 7eqtr4d 2213 . . 3  |-  ( -. 
ph  ->  if ( -. 
ph ,  A ,  B )  =  if ( ph ,  B ,  A ) )
95, 8jaoi 716 . 2  |-  ( (
ph  \/  -.  ph )  ->  if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A ) )
101, 9sylbi 121 1  |-  (DECID  ph  ->  if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 708  DECID wdc 834    = wceq 1353   ifcif 3534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-dc 835  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-if 3535
This theorem is referenced by:  lgsneg  14395  lgsdilem  14398
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