ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ifbieq12d Unicode version

Theorem ifbieq12d 3551
Description: Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
ifbieq12d.1  |-  ( ph  ->  ( ps  <->  ch )
)
ifbieq12d.2  |-  ( ph  ->  A  =  C )
ifbieq12d.3  |-  ( ph  ->  B  =  D )
Assertion
Ref Expression
ifbieq12d  |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  C ,  D )
)

Proof of Theorem ifbieq12d
StepHypRef Expression
1 ifbieq12d.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
21ifbid 3546 . 2  |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  A ,  B )
)
3 ifbieq12d.2 . . 3  |-  ( ph  ->  A  =  C )
4 ifbieq12d.3 . . 3  |-  ( ph  ->  B  =  D )
53, 4ifeq12d 3544 . 2  |-  ( ph  ->  if ( ch ,  A ,  B )  =  if ( ch ,  C ,  D )
)
62, 5eqtrd 2203 1  |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  C ,  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348   ifcif 3525
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-un 3125  df-if 3526
This theorem is referenced by:  updjudhcoinlf  7055  updjudhcoinrg  7056  omp1eom  7070  xaddval  9795  iseqf1olemqval  10436  iseqf1olemqk  10443  seq3f1olemqsum  10449  exp3val  10471  cvgratz  11488  eucalgval2  12000  ennnfonelemg  12351  ennnfonelem1  12355  ressid2  12470  ressval2  12471  lgsval  13664
  Copyright terms: Public domain W3C validator