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

Theorem anxordi 1378
Description: Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.)
Assertion
Ref Expression
anxordi  |-  ( (
ph  /\  ( ps  \/_ 
ch ) )  <->  ( ( ph  /\  ps )  \/_  ( ph  /\  ch )
) )

Proof of Theorem anxordi
StepHypRef Expression
1 simpl 108 . 2  |-  ( (
ph  /\  ( ps  \/_ 
ch ) )  ->  ph )
2 df-xor 1354 . . . 4  |-  ( ( ( ph  /\  ps )  \/_  ( ph  /\  ch ) )  <->  ( (
( ph  /\  ps )  \/  ( ph  /\  ch ) )  /\  -.  ( ( ph  /\  ps )  /\  ( ph  /\  ch ) ) ) )
32simplbi 272 . . 3  |-  ( ( ( ph  /\  ps )  \/_  ( ph  /\  ch ) )  ->  (
( ph  /\  ps )  \/  ( ph  /\  ch ) ) )
4 simpl 108 . . . 4  |-  ( (
ph  /\  ps )  ->  ph )
5 simpl 108 . . . 4  |-  ( (
ph  /\  ch )  ->  ph )
64, 5jaoi 705 . . 3  |-  ( ( ( ph  /\  ps )  \/  ( ph  /\ 
ch ) )  ->  ph )
73, 6syl 14 . 2  |-  ( ( ( ph  /\  ps )  \/_  ( ph  /\  ch ) )  ->  ph )
8 ibar 299 . . 3  |-  ( ph  ->  ( ( ps  \/_  ch )  <->  ( ph  /\  ( ps  \/_  ch )
) ) )
9 ibar 299 . . . 4  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
10 ibar 299 . . . 4  |-  ( ph  ->  ( ch  <->  ( ph  /\ 
ch ) ) )
119, 10xorbi12d 1360 . . 3  |-  ( ph  ->  ( ( ps  \/_  ch )  <->  ( ( ph  /\ 
ps )  \/_  ( ph  /\  ch ) ) ) )
128, 11bitr3d 189 . 2  |-  ( ph  ->  ( ( ph  /\  ( ps  \/_  ch )
)  <->  ( ( ph  /\ 
ps )  \/_  ( ph  /\  ch ) ) ) )
131, 7, 12pm5.21nii 693 1  |-  ( (
ph  /\  ( ps  \/_ 
ch ) )  <->  ( ( ph  /\  ps )  \/_  ( ph  /\  ch )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    \/ wo 697    \/_ wxo 1353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-xor 1354
This theorem is referenced by:  rpnegap  9481
  Copyright terms: Public domain W3C validator