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Theorem ifbieq12d2 23165
Description: Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.)
Hypotheses
Ref Expression
ifbieq12d2.1  |-  ( ph  ->  ( ps  <->  ch )
)
ifbieq12d2.2  |-  ( (
ph  /\  ps )  ->  A  =  C )
ifbieq12d2.3  |-  ( (
ph  /\  -.  ps )  ->  B  =  D )
Assertion
Ref Expression
ifbieq12d2  |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  C ,  D )
)

Proof of Theorem ifbieq12d2
StepHypRef Expression
1 exmid 404 . . . . 5  |-  ( ps  \/  -.  ps )
21a1i 10 . . . 4  |-  ( ph  ->  ( ps  \/  -.  ps ) )
3 ifbieq12d2.1 . . . . . . . . . 10  |-  ( ph  ->  ( ps  <->  ch )
)
4 iftrue 3584 . . . . . . . . . 10  |-  ( ch 
->  if ( ch ,  C ,  D )  =  C )
53, 4syl6bi 219 . . . . . . . . 9  |-  ( ph  ->  ( ps  ->  if ( ch ,  C ,  D )  =  C ) )
65imp 418 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  if ( ch ,  C ,  D )  =  C )
7 ifbieq12d2.2 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  A  =  C )
86, 7eqtr4d 2331 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  if ( ch ,  C ,  D )  =  A )
98ex 423 . . . . . 6  |-  ( ph  ->  ( ps  ->  if ( ch ,  C ,  D )  =  A ) )
109ancld 536 . . . . 5  |-  ( ph  ->  ( ps  ->  ( ps  /\  if ( ch ,  C ,  D
)  =  A ) ) )
113notbid 285 . . . . . . . . . 10  |-  ( ph  ->  ( -.  ps  <->  -.  ch )
)
12 iffalse 3585 . . . . . . . . . 10  |-  ( -. 
ch  ->  if ( ch ,  C ,  D
)  =  D )
1311, 12syl6bi 219 . . . . . . . . 9  |-  ( ph  ->  ( -.  ps  ->  if ( ch ,  C ,  D )  =  D ) )
1413imp 418 . . . . . . . 8  |-  ( (
ph  /\  -.  ps )  ->  if ( ch ,  C ,  D )  =  D )
15 ifbieq12d2.3 . . . . . . . 8  |-  ( (
ph  /\  -.  ps )  ->  B  =  D )
1614, 15eqtr4d 2331 . . . . . . 7  |-  ( (
ph  /\  -.  ps )  ->  if ( ch ,  C ,  D )  =  B )
1716ex 423 . . . . . 6  |-  ( ph  ->  ( -.  ps  ->  if ( ch ,  C ,  D )  =  B ) )
1817ancld 536 . . . . 5  |-  ( ph  ->  ( -.  ps  ->  ( -.  ps  /\  if ( ch ,  C ,  D )  =  B ) ) )
1910, 18orim12d 811 . . . 4  |-  ( ph  ->  ( ( ps  \/  -.  ps )  ->  (
( ps  /\  if ( ch ,  C ,  D )  =  A )  \/  ( -. 
ps  /\  if ( ch ,  C ,  D )  =  B ) ) ) )
202, 19mpd 14 . . 3  |-  ( ph  ->  ( ( ps  /\  if ( ch ,  C ,  D )  =  A )  \/  ( -. 
ps  /\  if ( ch ,  C ,  D )  =  B ) ) )
21 eqif 3611 . . 3  |-  ( if ( ch ,  C ,  D )  =  if ( ps ,  A ,  B )  <->  ( ( ps  /\  if ( ch ,  C ,  D
)  =  A )  \/  ( -.  ps  /\  if ( ch ,  C ,  D )  =  B ) ) )
2220, 21sylibr 203 . 2  |-  ( ph  ->  if ( ch ,  C ,  D )  =  if ( ps ,  A ,  B )
)
2322eqcomd 2301 1  |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  C ,  D )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632   ifcif 3578
This theorem is referenced by:  itgeq12dv  23211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-if 3579
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