Theorem pm4.78

Description: Implication distributes over disjunction. Theorem *4.78 of WhiteheadRussell p. 121. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 19-Nov-2012)

	Ref	Expression
Assertion	pm4.78	|- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) <-> ( ph -> ( ps \/ ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		orordi	|- ( ( -. ph \/ ( ps \/ ch ) ) <-> ( ( -. ph \/ ps ) \/ ( -. ph \/ ch ) ) )
2		imor	|- ( ( ph -> ( ps \/ ch ) ) <-> ( -. ph \/ ( ps \/ ch ) ) )
3		imor	|- ( ( ph -> ps ) <-> ( -. ph \/ ps ) )
4		imor	|- ( ( ph -> ch ) <-> ( -. ph \/ ch ) )
5	3 4	orbi12i	|- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) <-> ( ( -. ph \/ ps ) \/ ( -. ph \/ ch ) ) )
6	1 2 5	3bitr4ri	|- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) <-> ( ph -> ( ps \/ ch ) ) )