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Theorem bi2anan9r 844
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 843 . 2  |-  ( (
ph  /\  th )  ->  ( ( ps  /\  ta )  <->  ( ch  /\  et ) ) )
43ancoms 439 1  |-  ( ( th  /\  ph )  ->  ( ( ps  /\  ta )  <->  ( ch  /\  et ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358
This theorem is referenced by:  efrn2lp  4375  ltsosr  8716  seqf1olem2  11086  seqf1o  11087  pcval  12897  fneval  26287  prtlem5  26722  rmydioph  27107  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
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