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Theorem bi2anan9r 869
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 868 . 2  |-  ( (
ph  /\  th )  ->  ( ( ps  /\  ta )  <->  ( ch  /\  et ) ) )
43ancoms 453 1  |-  ( ( th  /\  ph )  ->  ( ( ps  /\  ta )  <->  ( ch  /\  et ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371
This theorem is referenced by:  efrn2lp  4702  ltsosr  9261  seqf1olem2  11846  seqf1o  11847  pcval  13911  fneval  28559  prtlem5  29001  rmydioph  29363  wepwsolem  29394  aomclem8  29414  usg2wlkeq  30289
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