Users' Mathboxes Mathbox for Jarvin Udandy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aiffbtbat Unicode version

Theorem aiffbtbat 27876
Description: Given a is equivalent to b, T. is equivalent to b. there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)
Hypotheses
Ref Expression
aiffbtbat.1  |-  ( ph  <->  ps )
aiffbtbat.2  |-  (  T.  <->  ps )
Assertion
Ref Expression
aiffbtbat  |-  ( ph  <->  T.  )

Proof of Theorem aiffbtbat
StepHypRef Expression
1 aiffbtbat.1 . . 3  |-  ( ph  <->  ps )
2 aiffbtbat.2 . . 3  |-  (  T.  <->  ps )
31, 2pm3.2i 441 . 2  |-  ( (
ph 
<->  ps )  /\  (  T. 
<->  ps ) )
4 biantr 897 . 2  |-  ( ( ( ph  <->  ps )  /\  (  T.  <->  ps )
)  ->  ( ph  <->  T.  ) )
53, 4ax-mp 8 1  |-  ( ph  <->  T.  )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    T. wtru 1307
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
  Copyright terms: Public domain W3C validator