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Theorem elim2if 23958
Description: Elimination of two conditional operators contained in a wff  ch. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Hypotheses
Ref Expression
elim2if.1  |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
)  =  A  -> 
( ch  <->  th )
)
elim2if.2  |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
)  =  B  -> 
( ch  <->  ta )
)
elim2if.3  |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
)  =  C  -> 
( ch  <->  et )
)
Assertion
Ref Expression
elim2if  |-  ( ch  <->  ( ( ph  /\  th )  \/  ( -.  ph 
/\  ( ( ps 
/\  ta )  \/  ( -.  ps  /\  et ) ) ) ) )

Proof of Theorem elim2if
StepHypRef Expression
1 exmid 405 . . 3  |-  ( ph  \/  -.  ph )
21biantrur 493 . 2  |-  ( ch  <->  ( ( ph  \/  -.  ph )  /\  ch )
)
3 andir 839 . 2  |-  ( ( ( ph  \/  -.  ph )  /\  ch )  <->  ( ( ph  /\  ch )  \/  ( -.  ph 
/\  ch ) ) )
4 iftrue 3705 . . . . 5  |-  ( ph  ->  if ( ph ,  A ,  if ( ps ,  B ,  C ) )  =  A )
5 elim2if.1 . . . . 5  |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
)  =  A  -> 
( ch  <->  th )
)
64, 5syl 16 . . . 4  |-  ( ph  ->  ( ch  <->  th )
)
76pm5.32i 619 . . 3  |-  ( (
ph  /\  ch )  <->  (
ph  /\  th )
)
8 iffalse 3706 . . . . . . 7  |-  ( -. 
ph  ->  if ( ph ,  A ,  if ( ps ,  B ,  C ) )  =  if ( ps ,  B ,  C )
)
98eqeq1d 2412 . . . . . 6  |-  ( -. 
ph  ->  ( if (
ph ,  A ,  if ( ps ,  B ,  C ) )  =  B  <->  if ( ps ,  B ,  C )  =  B ) )
10 elim2if.2 . . . . . 6  |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
)  =  B  -> 
( ch  <->  ta )
)
119, 10syl6bir 221 . . . . 5  |-  ( -. 
ph  ->  ( if ( ps ,  B ,  C )  =  B  ->  ( ch  <->  ta )
) )
128eqeq1d 2412 . . . . . 6  |-  ( -. 
ph  ->  ( if (
ph ,  A ,  if ( ps ,  B ,  C ) )  =  C  <->  if ( ps ,  B ,  C )  =  C ) )
13 elim2if.3 . . . . . 6  |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
)  =  C  -> 
( ch  <->  et )
)
1412, 13syl6bir 221 . . . . 5  |-  ( -. 
ph  ->  ( if ( ps ,  B ,  C )  =  C  ->  ( ch  <->  et )
) )
1511, 14elimifd 23957 . . . 4  |-  ( -. 
ph  ->  ( ch  <->  ( ( ps  /\  ta )  \/  ( -.  ps  /\  et ) ) ) )
1615pm5.32i 619 . . 3  |-  ( ( -.  ph  /\  ch )  <->  ( -.  ph  /\  (
( ps  /\  ta )  \/  ( -.  ps  /\  et ) ) ) )
177, 16orbi12i 508 . 2  |-  ( ( ( ph  /\  ch )  \/  ( -.  ph 
/\  ch ) )  <->  ( ( ph  /\  th )  \/  ( -.  ph  /\  ( ( ps  /\  ta )  \/  ( -.  ps  /\  et ) ) ) ) )
182, 3, 173bitri 263 1  |-  ( ch  <->  ( ( ph  /\  th )  \/  ( -.  ph 
/\  ( ( ps 
/\  ta )  \/  ( -.  ps  /\  et ) ) ) ) )
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:  elim2ifim  23959
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
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