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Metamath Proof Explorer

Theorem xordi

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 does hold in it. (Contributed by NM, 3-Oct-2008)

		Ref	Expression
	Assertion	xordi	\|- ( ( ph /\ -. ( ps <-> ch ) ) <-> -. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )

Proof

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

Step

Hyp

Ref

Expression

annim

 |-  ( ( ph /\ -. ( ps <-> ch ) ) <-> -. ( ph -> ( ps <-> ch ) ) )

pm5.32

 |-  ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )

1 2

xchbinx

 |-  ( ( ph /\ -. ( ps <-> ch ) ) <-> -. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )