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Theorem astbstanbst 27289
Description: Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for a and b is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)
Hypotheses
Ref Expression
astbstanbst.1  |-  ( ph  <->  T.  )
astbstanbst.2  |-  ( ps  <->  T.  )
Assertion
Ref Expression
astbstanbst  |-  ( (
ph  /\  ps )  <->  T.  )

Proof of Theorem astbstanbst
StepHypRef Expression
1 astbstanbst.1 . . . . 5  |-  ( ph  <->  T.  )
2 bi2 189 . . . . 5  |-  ( (
ph 
<->  T.  )  ->  (  T.  ->  ph ) )
31, 2ax-mp 8 . . . 4  |-  (  T. 
->  ph )
43trud 1314 . . 3  |-  ph
5 astbstanbst.2 . . . . 5  |-  ( ps  <->  T.  )
6 bi2 189 . . . . 5  |-  ( ( ps  <->  T.  )  ->  (  T.  ->  ps )
)
75, 6ax-mp 8 . . . 4  |-  (  T. 
->  ps )
87trud 1314 . . 3  |-  ps
94, 8pm3.2i 441 . 2  |-  ( ph  /\ 
ps )
109bitru 1317 1  |-  ( (
ph  /\  ps )  <->  T.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    T. wtru 1307
This theorem is referenced by:  dandysum2p2e4  27355
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  df-tru 1310
  Copyright terms: Public domain W3C validator