MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bi2bian9 Unicode version

Theorem bi2bian9 845
Description: Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)
Hypotheses
Ref Expression
bi2an9.1  |-  ( ph  ->  ( ps  <->  ch )
)
bi2an9.2  |-  ( th 
->  ( ta  <->  et )
)
Assertion
Ref Expression
bi2bian9  |-  ( (
ph  /\  th )  ->  ( ( ps  <->  ta )  <->  ( ch  <->  et ) ) )

Proof of Theorem bi2bian9
StepHypRef Expression
1 bi2an9.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
21adantr 451 . 2  |-  ( (
ph  /\  th )  ->  ( ps  <->  ch )
)
3 bi2an9.2 . . 3  |-  ( th 
->  ( ta  <->  et )
)
43adantl 452 . 2  |-  ( (
ph  /\  th )  ->  ( ta  <->  et )
)
52, 4bibi12d 312 1  |-  ( (
ph  /\  th )  ->  ( ( ps  <->  ta )  <->  ( ch  <->  et ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358
This theorem is referenced by:  uzindOLD  10106  wepwsolem  27138  aomclem8  27159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360
  Copyright terms: Public domain W3C validator