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Theorem anxordi 1332
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 107 . 2  |-  ( (
ph  /\  ( ps  \/_ 
ch ) )  ->  ph )
2 df-xor 1308 . . . 4  |-  ( ( ( ph  /\  ps )  \/_  ( ph  /\  ch ) )  <->  ( (
( ph  /\  ps )  \/  ( ph  /\  ch ) )  /\  -.  ( ( ph  /\  ps )  /\  ( ph  /\  ch ) ) ) )
32simplbi 268 . . 3  |-  ( ( ( ph  /\  ps )  \/_  ( ph  /\  ch ) )  ->  (
( ph  /\  ps )  \/  ( ph  /\  ch ) ) )
4 simpl 107 . . . 4  |-  ( (
ph  /\  ps )  ->  ph )
5 simpl 107 . . . 4  |-  ( (
ph  /\  ch )  ->  ph )
64, 5jaoi 669 . . 3  |-  ( ( ( ph  /\  ps )  \/  ( ph  /\ 
ch ) )  ->  ph )
73, 6syl 14 . 2  |-  ( ( ( ph  /\  ps )  \/_  ( ph  /\  ch ) )  ->  ph )
8 ibar 295 . . 3  |-  ( ph  ->  ( ( ps  \/_  ch )  <->  ( ph  /\  ( ps  \/_  ch )
) ) )
9 ibar 295 . . . 4  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
10 ibar 295 . . . 4  |-  ( ph  ->  ( ch  <->  ( ph  /\ 
ch ) ) )
119, 10xorbi12d 1314 . . 3  |-  ( ph  ->  ( ( ps  \/_  ch )  <->  ( ( ph  /\ 
ps )  \/_  ( ph  /\  ch ) ) ) )
128, 11bitr3d 188 . 2  |-  ( ph  ->  ( ( ph  /\  ( ps  \/_  ch )
)  <->  ( ( ph  /\ 
ps )  \/_  ( ph  /\  ch ) ) ) )
131, 7, 12pm5.21nii 653 1  |-  ( (
ph  /\  ( ps  \/_ 
ch ) )  <->  ( ( ph  /\  ps )  \/_  ( ph  /\  ch )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    <-> wb 103    \/ wo 662    \/_ wxo 1307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-xor 1308
This theorem is referenced by:  rpnegap  8847
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