Theorem pm4.78 606

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.)

Assertion

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

Proof of Theorem pm4.78

Step	Hyp	Ref	Expression
1		orordi 552	. 2 |- ( ( -. ph \/ ( ps \/ ch ) ) <-> ( ( -. ph \/ ps ) \/ ( -. ph \/ ch ) ) )
2		imor 428	. 2 |- ( ( ph -> ( ps \/ ch ) ) <-> ( -. ph \/ ( ps \/ ch ) ) )
3		imor 428	. . 3 |- ( ( ph -> ps ) <-> ( -. ph \/ ps ) )
4		imor 428	. . 3 |- ( ( ph -> ch ) <-> ( -. ph \/ ch ) )
5	3, 4	orbi12i 543	. 2 |- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) <-> ( ( -. ph \/ ps ) \/ ( -. ph \/ ch ) ) )
6	1, 2, 5	3bitr4ri 293	1 |- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) <-> ( ph -> ( ps \/ ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383

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-or 385

This theorem is referenced by: (None)