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Theorem elimif 3930
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 3904 . . 3  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
2 elimif.1 . . 3  |-  ( if ( ph ,  A ,  B )  =  A  ->  ( ps  <->  ch )
)
31, 2syl 16 . 2  |-  ( ph  ->  ( ps  <->  ch )
)
4 iffalse 3906 . . 3  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
5 elimif.2 . . 3  |-  ( if ( ph ,  A ,  B )  =  B  ->  ( ps  <->  th )
)
64, 5syl 16 . 2  |-  ( -. 
ph  ->  ( ps  <->  th )
)
73, 6cases 962 1  |-  ( ps  <->  ( ( ph  /\  ch )  \/  ( -.  ph 
/\  th ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370   ifcif 3898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-if 3899
This theorem is referenced by:  eqif  3934  elif  3936  ifel  3937  ftc1anclem5  28618
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