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Theorem elimif 3973
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 3945 . . 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 3948 . . 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 969 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 1379   ifcif 3939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-if 3940
This theorem is referenced by:  eqif  3977  elif  3979  ifel  3980  ftc1anclem5  29671
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