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Theorem iffalse 3477
Description: Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
Assertion
Ref Expression
iffalse  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )

Proof of Theorem iffalse
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dedlemb 954 . . 3  |-  ( -. 
ph  ->  ( x  e.  B  <->  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) ) )
21abbi2dv 2256 . 2  |-  ( -. 
ph  ->  B  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) } )
3 df-if 3470 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
42, 3syl6reqr 2189 1  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 697    = wceq 1331    e. wcel 1480   {cab 2123   ifcif 3469
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 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-if 3470
This theorem is referenced by:  iffalsei  3478  iffalsed  3479  ifnefalse  3480  ifsbdc  3481  ifcldadc  3496  ifeq1dadc  3497  ifbothdadc  3498  ifbothdc  3499  ifiddc  3500  ifcldcd  3502  ifandc  3503  fidifsnen  6757  nnnninf  7016  uzin  9351  modifeq2int  10152  bcval  10488  bcval3  10490  sumrbdclem  11138  fsum3cvg  11139  summodclem2a  11143  sumsplitdc  11194  prodrbdclem  11333  fproddccvg  11334  flodddiv4  11620  gcdn0val  11639  dfgcd2  11691  lcmn0val  11736  unct  11943
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