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Theorem ifiddc 3567
Description: Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
Assertion
Ref Expression
ifiddc  |-  (DECID  ph  ->  if ( ph ,  A ,  A )  =  A )

Proof of Theorem ifiddc
StepHypRef Expression
1 exmiddc 836 . 2  |-  (DECID  ph  ->  (
ph  \/  -.  ph )
)
2 iftrue 3539 . . 3  |-  ( ph  ->  if ( ph ,  A ,  A )  =  A )
3 iffalse 3542 . . 3  |-  ( -. 
ph  ->  if ( ph ,  A ,  A )  =  A )
42, 3jaoi 716 . 2  |-  ( (
ph  \/  -.  ph )  ->  if ( ph ,  A ,  A )  =  A )
51, 4syl 14 1  |-  (DECID  ph  ->  if ( ph ,  A ,  A )  =  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-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:  xaddpnf1  9830  xaddmnf1  9832  isumz  11378  prod1dc  11575  1arithlem4  12344  lgsval2lem  14075  lgsdilem2  14101
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