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Theorem xordi 904
Description: Conjunction distributes over exclusive-or, using  -.  ( ph  <->  ps ) to express exclusive-or. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. This is not necessarily true in intuitionistic logic, though anxordi 1419 does hold in it. (Contributed by NM, 3-Oct-2008.)
Assertion
Ref Expression
xordi  |-  ( (
ph  /\  -.  ( ps 
<->  ch ) )  <->  -.  (
( ph  /\  ps )  <->  (
ph  /\  ch )
) )

Proof of Theorem xordi
StepHypRef Expression
1 annim 427 . 2  |-  ( (
ph  /\  -.  ( ps 
<->  ch ) )  <->  -.  ( ph  ->  ( ps  <->  ch )
) )
2 pm5.32 641 . 2  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  /\ 
ps )  <->  ( ph  /\ 
ch ) ) )
31, 2xchbinx 312 1  |-  ( (
ph  /\  -.  ( ps 
<->  ch ) )  <->  -.  (
( ph  /\  ps )  <->  (
ph  /\  ch )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 189  df-an 373
This theorem is referenced by:  anxordi  1419
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