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Theorem mpbir2and 862
Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
Hypotheses
Ref Expression
mpbir2and.1  |-  ( ph  ->  ch )
mpbir2and.2  |-  ( ph  ->  th )
mpbir2and.3  |-  ( ph  ->  ( ps  <->  ( ch  /\ 
th ) ) )
Assertion
Ref Expression
mpbir2and  |-  ( ph  ->  ps )

Proof of Theorem mpbir2and
StepHypRef Expression
1 mpbir2and.1 . . 3  |-  ( ph  ->  ch )
2 mpbir2and.2 . . 3  |-  ( ph  ->  th )
31, 2jca 294 . 2  |-  ( ph  ->  ( ch  /\  th ) )
4 mpbir2and.3 . 2  |-  ( ph  ->  ( ps  <->  ( ch  /\ 
th ) ) )
53, 4mpbird 160 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  nqnq0pi  6594  genpassg  6682  addnqpr  6717  mulnqpr  6733  distrprg  6744  1idpr  6748  ltexpri  6769  recexprlemex  6793  aptipr  6797  cauappcvgprlemladd  6814  add20  7543  inelr  7649  recgt0  7891  prodgt0  7893  squeeze0  7945  eluzadd  8597  eluzsub  8598  xrre  8834  xrre3  8836  elioc2  8906  elico2  8907  elicc2  8908  elfz1eq  9001  fztri3or  9005  fznatpl1  9040  nn0fz0  9080  fzctr  9093  fzo1fzo0n0  9141  fzoaddel  9150  qbtwnz  9208  flid  9234  flqaddz  9247  flqdiv  9271  modqid  9299  frec2uzf1od  9356  expival  9422  ibcval5  9631  abs2difabs  9935  fzomaxdiflem  9939  icodiamlt  10007  pw2dvdseu  10256  nn0seqcvgd  10263
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