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Theorem elimif 3918
Description: Elimination of a conditional operator contained in a wff  ps. (Contributed by NM, 15-Feb-2005.) (Proof shortened by NM, 25-Apr-2019.)
Hypotheses
Ref Expression
elimif.1  |-  ( if ( ph ,  A ,  B )  =  A  ->  ( ps  <->  ch )
)
elimif.2  |-  ( if ( ph ,  A ,  B )  =  B  ->  ( ps  <->  th )
)
Assertion
Ref Expression
elimif  |-  ( ps  <->  ( ( ph  /\  ch )  \/  ( -.  ph 
/\  th ) ) )

Proof of Theorem elimif
StepHypRef Expression
1 iftrue 3890 . . 3  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
2 elimif.1 . . 3  |-  ( if ( ph ,  A ,  B )  =  A  ->  ( ps  <->  ch )
)
31, 2syl 17 . 2  |-  ( ph  ->  ( ps  <->  ch )
)
4 iffalse 3893 . . 3  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
5 elimif.2 . . 3  |-  ( if ( ph ,  A ,  B )  =  B  ->  ( ps  <->  th )
)
64, 5syl 17 . 2  |-  ( -. 
ph  ->  ( ps  <->  th )
)
73, 6cases 971 1  |-  ( ps  <->  ( ( ph  /\  ch )  \/  ( -.  ph 
/\  th ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1405   ifcif 3884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-if 3885
This theorem is referenced by:  eqif  3922  elif  3924  ifel  3925  ftc1anclem5  31447
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