MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfbi2 Unicode version

Theorem dfbi2 612
Description: A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
dfbi2  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )

Proof of Theorem dfbi2
StepHypRef Expression
1 dfbi1 186 . 2  |-  ( (
ph 
<->  ps )  <->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )
2 df-an 362 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  <->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )
31, 2bitr4i 245 1  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360
This theorem is referenced by:  dfbi  613  pm4.71  614  pm5.17  863  xor  866  albiim  1612  nfbi  1738  sbbi  1964  cleqh  2355  ralbiim  2655  reu8  2936  sseq2  3175  soeq2  4306  fun11  5253  dffo3  5609  isnsg2  14610  axextprim  23420  biimpexp  23443  axextndbi  23531  aibandbiaiffaiffb  27211  aibandbiaiaiffb  27212  aibnbna  27223  equextvK  28323
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-an 362
  Copyright terms: Public domain W3C validator