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

Theorem anxordi 1375
Description: Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 893 does not necessarily hold, in part because the usual definition of xor is subtly different in intuitionistic logic. (Contributed by David A. Wheeler, 7-Oct-2018.)
Assertion
Ref Expression
anxordi  |-  ( (
ph  /\  ( ps  \/_ 
ch ) )  <->  ( ( ph  /\  ps )  \/_  ( ph  /\  ch )
) )

Proof of Theorem anxordi
StepHypRef Expression
1 xordi 893 . 2  |-  ( (
ph  /\  -.  ( ps 
<->  ch ) )  <->  -.  (
( ph  /\  ps )  <->  (
ph  /\  ch )
) )
2 df-xor 1361 . . 3  |-  ( ( ps  \/_  ch )  <->  -.  ( ps  <->  ch )
)
32anbi2i 694 . 2  |-  ( (
ph  /\  ( ps  \/_ 
ch ) )  <->  ( ph  /\ 
-.  ( ps  <->  ch )
) )
4 df-xor 1361 . 2  |-  ( ( ( ph  /\  ps )  \/_  ( ph  /\  ch ) )  <->  -.  (
( ph  /\  ps )  <->  (
ph  /\  ch )
) )
51, 3, 43bitr4i 277 1  |-  ( (
ph  /\  ( ps  \/_ 
ch ) )  <->  ( ( ph  /\  ps )  \/_  ( ph  /\  ch )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    \/_ wxo 1360
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 185  df-an 371  df-xor 1361
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator