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Theorem ifiddc 3559
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 831 . 2  |-  (DECID  ph  ->  (
ph  \/  -.  ph )
)
2 iftrue 3531 . . 3  |-  ( ph  ->  if ( ph ,  A ,  A )  =  A )
3 iffalse 3534 . . 3  |-  ( -. 
ph  ->  if ( ph ,  A ,  A )  =  A )
42, 3jaoi 711 . 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 703  DECID wdc 829    = wceq 1348   ifcif 3526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-dc 830  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-if 3527
This theorem is referenced by:  xaddpnf1  9803  xaddmnf1  9805  isumz  11352  prod1dc  11549  1arithlem4  12318  lgsval2lem  13705  lgsdilem2  13731
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