ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  bi2anan9r Unicode version

Theorem bi2anan9r 539
Description: Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.)
Hypotheses
Ref Expression
bi2an9.1  |-  ( ph  ->  ( ps  <->  ch )
)
bi2an9.2  |-  ( th 
->  ( ta  <->  et )
)
Assertion
Ref Expression
bi2anan9r  |-  ( ( th  /\  ph )  ->  ( ( ps  /\  ta )  <->  ( ch  /\  et ) ) )

Proof of Theorem bi2anan9r
StepHypRef Expression
1 bi2an9.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
2 bi2an9.2 . . 3  |-  ( th 
->  ( ta  <->  et )
)
31, 2bi2anan9 538 . 2  |-  ( (
ph  /\  th )  ->  ( ( ps  /\  ta )  <->  ( ch  /\  et ) ) )
43ancoms 255 1  |-  ( ( th  /\  ph )  ->  ( ( ps  /\  ta )  <->  ( ch  /\  et ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator