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Theorem elimifd 23957
Description: Elimination of a conditional operator contained in a wff  ch. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Hypotheses
Ref Expression
elimifd.1  |-  ( ph  ->  ( if ( ps ,  A ,  B
)  =  A  -> 
( ch  <->  th )
) )
elimifd.2  |-  ( ph  ->  ( if ( ps ,  A ,  B
)  =  B  -> 
( ch  <->  ta )
) )
Assertion
Ref Expression
elimifd  |-  ( ph  ->  ( ch  <->  ( ( ps  /\  th )  \/  ( -.  ps  /\  ta ) ) ) )

Proof of Theorem elimifd
StepHypRef Expression
1 exmid 405 . . . 4  |-  ( ps  \/  -.  ps )
21biantrur 493 . . 3  |-  ( ch  <->  ( ( ps  \/  -.  ps )  /\  ch )
)
32a1i 11 . 2  |-  ( ph  ->  ( ch  <->  ( ( ps  \/  -.  ps )  /\  ch ) ) )
4 andir 839 . . 3  |-  ( ( ( ps  \/  -.  ps )  /\  ch )  <->  ( ( ps  /\  ch )  \/  ( -.  ps  /\  ch ) ) )
54a1i 11 . 2  |-  ( ph  ->  ( ( ( ps  \/  -.  ps )  /\  ch )  <->  ( ( ps  /\  ch )  \/  ( -.  ps  /\  ch ) ) ) )
6 iftrue 3705 . . . . 5  |-  ( ps 
->  if ( ps ,  A ,  B )  =  A )
7 elimifd.1 . . . . 5  |-  ( ph  ->  ( if ( ps ,  A ,  B
)  =  A  -> 
( ch  <->  th )
) )
86, 7syl5 30 . . . 4  |-  ( ph  ->  ( ps  ->  ( ch 
<->  th ) ) )
98pm5.32d 621 . . 3  |-  ( ph  ->  ( ( ps  /\  ch )  <->  ( ps  /\  th ) ) )
10 iffalse 3706 . . . . 5  |-  ( -. 
ps  ->  if ( ps ,  A ,  B
)  =  B )
11 elimifd.2 . . . . 5  |-  ( ph  ->  ( if ( ps ,  A ,  B
)  =  B  -> 
( ch  <->  ta )
) )
1210, 11syl5 30 . . . 4  |-  ( ph  ->  ( -.  ps  ->  ( ch  <->  ta ) ) )
1312pm5.32d 621 . . 3  |-  ( ph  ->  ( ( -.  ps  /\ 
ch )  <->  ( -.  ps  /\  ta ) ) )
149, 13orbi12d 691 . 2  |-  ( ph  ->  ( ( ( ps 
/\  ch )  \/  ( -.  ps  /\  ch )
)  <->  ( ( ps 
/\  th )  \/  ( -.  ps  /\  ta )
) ) )
153, 5, 143bitrd 271 1  |-  ( ph  ->  ( ch  <->  ( ( ps  /\  th )  \/  ( -.  ps  /\  ta ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649   ifcif 3699
This theorem is referenced by:  elim2if  23958
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-if 3700
  Copyright terms: Public domain W3C validator