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Theorem elimif 3594
Description: Elimination of a conditional operator contained in a wff  ps. (Contributed by NM, 15-Feb-2005.)
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 exmid 404 . . 3  |-  ( ph  \/  -.  ph )
21biantrur 492 . 2  |-  ( ps  <->  ( ( ph  \/  -.  ph )  /\  ps )
)
3 andir 838 . 2  |-  ( ( ( ph  \/  -.  ph )  /\  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ps ) ) )
4 iftrue 3571 . . . . 5  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
5 elimif.1 . . . . 5  |-  ( if ( ph ,  A ,  B )  =  A  ->  ( ps  <->  ch )
)
64, 5syl 15 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
76pm5.32i 618 . . 3  |-  ( (
ph  /\  ps )  <->  (
ph  /\  ch )
)
8 iffalse 3572 . . . . 5  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
9 elimif.2 . . . . 5  |-  ( if ( ph ,  A ,  B )  =  B  ->  ( ps  <->  th )
)
108, 9syl 15 . . . 4  |-  ( -. 
ph  ->  ( ps  <->  th )
)
1110pm5.32i 618 . . 3  |-  ( ( -.  ph  /\  ps )  <->  ( -.  ph  /\  th )
)
127, 11orbi12i 507 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ps ) )  <->  ( ( ph  /\  ch )  \/  ( -.  ph  /\  th ) ) )
132, 3, 123bitri 262 1  |-  ( ps  <->  ( ( ph  /\  ch )  \/  ( -.  ph 
/\  th ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623   ifcif 3565
This theorem is referenced by:  eqif  3598  elif  3599  ifel  3600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-if 3566
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