ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ifbid Unicode version

Theorem ifbid 3493
Description: Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
Hypothesis
Ref Expression
ifbid.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
ifbid  |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  A ,  B )
)

Proof of Theorem ifbid
StepHypRef Expression
1 ifbid.1 . 2  |-  ( ph  ->  ( ps  <->  ch )
)
2 ifbi 3492 . 2  |-  ( ( ps  <->  ch )  ->  if ( ps ,  A ,  B )  =  if ( ch ,  A ,  B ) )
31, 2syl 14 1  |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  A ,  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331   ifcif 3474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-if 3475
This theorem is referenced by:  ifbieq1d  3494  ifbieq2d  3496  ifbieq12d  3498  ifandc  3508  fodjum  7018  fodju0  7019  fodjuomni  7021  nnnninf  7023  fodjumkv  7034  xaddval  9635  0tonninf  10219  1tonninf  10220  sumeq1  11131  summodc  11159  zsumdc  11160  fsum3  11163  isumss  11167  sumsplitdc  11208  prodeq1f  11328  subctctexmid  13210  nninfalllemn  13216  nninfalllem1  13217  nninfsellemdc  13220  nninfself  13223  nninfsellemeq  13224  nninfsellemqall  13225  nninfsellemeqinf  13226  nninfomni  13229  nninffeq  13230
  Copyright terms: Public domain W3C validator