ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm5.1im Unicode version

Theorem pm5.1im 171
Description: Two propositions are equivalent if they are both true. Closed form of 2th 172. Equivalent to a bi1 116-like version of the xor-connective. This theorem stays true, no matter how you permute its operands. This is evident from its sharper version 
( ph  <->  ( ps  <->  ( ph  <->  ps ) ) ). (Contributed by Wolf Lammen, 12-May-2013.)
Assertion
Ref Expression
pm5.1im  |-  ( ph  ->  ( ps  ->  ( ph 
<->  ps ) ) )

Proof of Theorem pm5.1im
StepHypRef Expression
1 ax-1 5 . 2  |-  ( ps 
->  ( ph  ->  ps ) )
2 ax-1 5 . 2  |-  ( ph  ->  ( ps  ->  ph )
)
31, 2impbid21d 126 1  |-  ( ph  ->  ( ps  ->  ( ph 
<->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  2thd  173  pm5.501  242  dcextest  4386  snexxph  6638
  Copyright terms: Public domain W3C validator