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Theorem ifbieq12d2 26040
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 415 . . . 4  |-  ( ps  \/  -.  ps )
2 ifbieq12d2.1 . . . . . . . . . 10  |-  ( ph  ->  ( ps  <->  ch )
)
3 iftrue 3897 . . . . . . . . . 10  |-  ( ch 
->  if ( ch ,  C ,  D )  =  C )
42, 3syl6bi 228 . . . . . . . . 9  |-  ( ph  ->  ( ps  ->  if ( ch ,  C ,  D )  =  C ) )
54imp 429 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  if ( ch ,  C ,  D )  =  C )
6 ifbieq12d2.2 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  A  =  C )
75, 6eqtr4d 2495 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  if ( ch ,  C ,  D )  =  A )
87ex 434 . . . . . 6  |-  ( ph  ->  ( ps  ->  if ( ch ,  C ,  D )  =  A ) )
98ancld 553 . . . . 5  |-  ( ph  ->  ( ps  ->  ( ps  /\  if ( ch ,  C ,  D
)  =  A ) ) )
102notbid 294 . . . . . . . . . 10  |-  ( ph  ->  ( -.  ps  <->  -.  ch )
)
11 iffalse 3899 . . . . . . . . . 10  |-  ( -. 
ch  ->  if ( ch ,  C ,  D
)  =  D )
1210, 11syl6bi 228 . . . . . . . . 9  |-  ( ph  ->  ( -.  ps  ->  if ( ch ,  C ,  D )  =  D ) )
1312imp 429 . . . . . . . 8  |-  ( (
ph  /\  -.  ps )  ->  if ( ch ,  C ,  D )  =  D )
14 ifbieq12d2.3 . . . . . . . 8  |-  ( (
ph  /\  -.  ps )  ->  B  =  D )
1513, 14eqtr4d 2495 . . . . . . 7  |-  ( (
ph  /\  -.  ps )  ->  if ( ch ,  C ,  D )  =  B )
1615ex 434 . . . . . 6  |-  ( ph  ->  ( -.  ps  ->  if ( ch ,  C ,  D )  =  B ) )
1716ancld 553 . . . . 5  |-  ( ph  ->  ( -.  ps  ->  ( -.  ps  /\  if ( ch ,  C ,  D )  =  B ) ) )
189, 17orim12d 834 . . . 4  |-  ( ph  ->  ( ( ps  \/  -.  ps )  ->  (
( ps  /\  if ( ch ,  C ,  D )  =  A )  \/  ( -. 
ps  /\  if ( ch ,  C ,  D )  =  B ) ) ) )
191, 18mpi 17 . . 3  |-  ( ph  ->  ( ( ps  /\  if ( ch ,  C ,  D )  =  A )  \/  ( -. 
ps  /\  if ( ch ,  C ,  D )  =  B ) ) )
20 eqif 3927 . . 3  |-  ( if ( ch ,  C ,  D )  =  if ( ps ,  A ,  B )  <->  ( ( ps  /\  if ( ch ,  C ,  D
)  =  A )  \/  ( -.  ps  /\  if ( ch ,  C ,  D )  =  B ) ) )
2119, 20sylibr 212 . 2  |-  ( ph  ->  if ( ch ,  C ,  D )  =  if ( ps ,  A ,  B )
)
2221eqcomd 2459 1  |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  C ,  D )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370   ifcif 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-if 3892
This theorem is referenced by:  itgeq12dv  26848  sgnneg  27059  ofccat  27077
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