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Theorem biadan2 449
Description: Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.)
Hypotheses
Ref Expression
biadan2.1  |-  ( ph  ->  ps )
biadan2.2  |-  ( ps 
->  ( ph  <->  ch )
)
Assertion
Ref Expression
biadan2  |-  ( ph  <->  ( ps  /\  ch )
)

Proof of Theorem biadan2
StepHypRef Expression
1 biadan2.1 . . 3  |-  ( ph  ->  ps )
21pm4.71ri 387 . 2  |-  ( ph  <->  ( ps  /\  ph )
)
3 biadan2.2 . . 3  |-  ( ps 
->  ( ph  <->  ch )
)
43pm5.32i 447 . 2  |-  ( ( ps  /\  ph )  <->  ( ps  /\  ch )
)
52, 4bitri 183 1  |-  ( ph  <->  ( ps  /\  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  elab4g  2804  elpwb  3488  ssdifsn  3619  brab2a  4560  brab2ga  4582  elovmpo  5937  eqop2  6042  elnnnn0  8974  elixx3g  9635  elfzo2  9878  1nprm  11702
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