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Theorem ifbieq12d2 24003
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 406 . . . 4  |-  ( ps  \/  -.  ps )
2 ifbieq12d2.1 . . . . . . . . . 10  |-  ( ph  ->  ( ps  <->  ch )
)
3 iftrue 3746 . . . . . . . . . 10  |-  ( ch 
->  if ( ch ,  C ,  D )  =  C )
42, 3syl6bi 221 . . . . . . . . 9  |-  ( ph  ->  ( ps  ->  if ( ch ,  C ,  D )  =  C ) )
54imp 420 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  if ( ch ,  C ,  D )  =  C )
6 ifbieq12d2.2 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  A  =  C )
75, 6eqtr4d 2472 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  if ( ch ,  C ,  D )  =  A )
87ex 425 . . . . . 6  |-  ( ph  ->  ( ps  ->  if ( ch ,  C ,  D )  =  A ) )
98ancld 538 . . . . 5  |-  ( ph  ->  ( ps  ->  ( ps  /\  if ( ch ,  C ,  D
)  =  A ) ) )
102notbid 287 . . . . . . . . . 10  |-  ( ph  ->  ( -.  ps  <->  -.  ch )
)
11 iffalse 3747 . . . . . . . . . 10  |-  ( -. 
ch  ->  if ( ch ,  C ,  D
)  =  D )
1210, 11syl6bi 221 . . . . . . . . 9  |-  ( ph  ->  ( -.  ps  ->  if ( ch ,  C ,  D )  =  D ) )
1312imp 420 . . . . . . . 8  |-  ( (
ph  /\  -.  ps )  ->  if ( ch ,  C ,  D )  =  D )
14 ifbieq12d2.3 . . . . . . . 8  |-  ( (
ph  /\  -.  ps )  ->  B  =  D )
1513, 14eqtr4d 2472 . . . . . . 7  |-  ( (
ph  /\  -.  ps )  ->  if ( ch ,  C ,  D )  =  B )
1615ex 425 . . . . . 6  |-  ( ph  ->  ( -.  ps  ->  if ( ch ,  C ,  D )  =  B ) )
1716ancld 538 . . . . 5  |-  ( ph  ->  ( -.  ps  ->  ( -.  ps  /\  if ( ch ,  C ,  D )  =  B ) ) )
189, 17orim12d 813 . . . 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 3773 . . 3  |-  ( if ( ch ,  C ,  D )  =  if ( ps ,  A ,  B )  <->  ( ( ps  /\  if ( ch ,  C ,  D
)  =  A )  \/  ( -.  ps  /\  if ( ch ,  C ,  D )  =  B ) ) )
2119, 20sylibr 205 . 2  |-  ( ph  ->  if ( ch ,  C ,  D )  =  if ( ps ,  A ,  B )
)
2221eqcomd 2442 1  |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  C ,  D )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653   ifcif 3740
This theorem is referenced by:  itgeq12dv  24642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-if 3741
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