Theorem xordi 937

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 1479 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

Step	Hyp	Ref	Expression
1		annim 441	. 2 |- ( ( ph /\ -. ( ps <-> ch ) ) <-> -. ( ph -> ( ps <-> ch ) ) )
2		pm5.32 668	. 2 |- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )
3	1, 2	xchbinx 324	1 |- ( ( ph /\ -. ( ps <-> ch ) ) <-> -. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384

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 197  df-an 386

This theorem is referenced by:  anxordi  1479