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Theorem elim2ifim 27125
Description: Elimination of two conditional operators for an implication. (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 )
)
elim2ifim.1  |-  ( ph  ->  th )
elim2ifim.2  |-  ( ( -.  ph  /\  ps )  ->  ta )
elim2ifim.3  |-  ( ( -.  ph  /\  -.  ps )  ->  et )
Assertion
Ref Expression
elim2ifim  |-  ch

Proof of Theorem elim2ifim
StepHypRef Expression
1 exmid 415 . . 3  |-  ( ph  \/  -.  ph )
2 elim2ifim.1 . . . . 5  |-  ( ph  ->  th )
32ancli 551 . . . 4  |-  ( ph  ->  ( ph  /\  th ) )
4 pm4.42 958 . . . . . 6  |-  ( -. 
ph 
<->  ( ( -.  ph  /\ 
ps )  \/  ( -.  ph  /\  -.  ps ) ) )
5 elim2ifim.2 . . . . . . . . . 10  |-  ( ( -.  ph  /\  ps )  ->  ta )
65ex 434 . . . . . . . . 9  |-  ( -. 
ph  ->  ( ps  ->  ta ) )
76ancld 553 . . . . . . . 8  |-  ( -. 
ph  ->  ( ps  ->  ( ps  /\  ta )
) )
87imp 429 . . . . . . 7  |-  ( ( -.  ph  /\  ps )  ->  ( ps  /\  ta ) )
9 elim2ifim.3 . . . . . . . . . 10  |-  ( ( -.  ph  /\  -.  ps )  ->  et )
109ex 434 . . . . . . . . 9  |-  ( -. 
ph  ->  ( -.  ps  ->  et ) )
1110ancld 553 . . . . . . . 8  |-  ( -. 
ph  ->  ( -.  ps  ->  ( -.  ps  /\  et ) ) )
1211imp 429 . . . . . . 7  |-  ( ( -.  ph  /\  -.  ps )  ->  ( -.  ps  /\  et ) )
138, 12orim12i 516 . . . . . 6  |-  ( ( ( -.  ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) )  ->  (
( ps  /\  ta )  \/  ( -.  ps  /\  et ) ) )
144, 13sylbi 195 . . . . 5  |-  ( -. 
ph  ->  ( ( ps 
/\  ta )  \/  ( -.  ps  /\  et ) ) )
1514ancli 551 . . . 4  |-  ( -. 
ph  ->  ( -.  ph  /\  ( ( ps  /\  ta )  \/  ( -.  ps  /\  et ) ) ) )
163, 15orim12i 516 . . 3  |-  ( (
ph  \/  -.  ph )  ->  ( ( ph  /\  th )  \/  ( -. 
ph  /\  ( ( ps  /\  ta )  \/  ( -.  ps  /\  et ) ) ) ) )
171, 16ax-mp 5 . 2  |-  ( (
ph  /\  th )  \/  ( -.  ph  /\  ( ( ps  /\  ta )  \/  ( -.  ps  /\  et ) ) ) )
18 elim2if.1 . . 3  |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
)  =  A  -> 
( ch  <->  th )
)
19 elim2if.2 . . 3  |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
)  =  B  -> 
( ch  <->  ta )
)
20 elim2if.3 . . 3  |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
)  =  C  -> 
( ch  <->  et )
)
2118, 19, 20elim2if 27124 . 2  |-  ( ch  <->  ( ( ph  /\  th )  \/  ( -.  ph 
/\  ( ( ps 
/\  ta )  \/  ( -.  ps  /\  et ) ) ) ) )
2217, 21mpbir 209 1  |-  ch
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: (None)
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