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Theorem elimif 3713
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 405 . . 3  |-  ( ph  \/  -.  ph )
21biantrur 493 . 2  |-  ( ps  <->  ( ( ph  \/  -.  ph )  /\  ps )
)
3 andir 839 . 2  |-  ( ( ( ph  \/  -.  ph )  /\  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ps ) ) )
4 iftrue 3690 . . . . 5  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
5 elimif.1 . . . . 5  |-  ( if ( ph ,  A ,  B )  =  A  ->  ( ps  <->  ch )
)
64, 5syl 16 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
76pm5.32i 619 . . 3  |-  ( (
ph  /\  ps )  <->  (
ph  /\  ch )
)
8 iffalse 3691 . . . . 5  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
9 elimif.2 . . . . 5  |-  ( if ( ph ,  A ,  B )  =  B  ->  ( ps  <->  th )
)
108, 9syl 16 . . . 4  |-  ( -. 
ph  ->  ( ps  <->  th )
)
1110pm5.32i 619 . . 3  |-  ( ( -.  ph  /\  ps )  <->  ( -.  ph  /\  th )
)
127, 11orbi12i 508 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ps ) )  <->  ( ( ph  /\  ch )  \/  ( -.  ph  /\  th ) ) )
132, 3, 123bitri 263 1  |-  ( ps  <->  ( ( ph  /\  ch )  \/  ( -.  ph 
/\  th ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649   ifcif 3684
This theorem is referenced by:  eqif  3717  elif  3718  ifel  3719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-if 3685
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