Theorem pm4.78i 772

Description: Implication distributes over disjunction. One direction of Theorem *4.78 of [WhiteheadRussell] p. 121. The converse holds in classical logic. (Contributed by Jim Kingdon, 15-Jan-2018.)

Assertion

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

Proof of Theorem pm4.78i

Step	Hyp	Ref	Expression
1		orc 702	. . 3 |- ( ps -> ( ps \/ ch ) )
2	1	imim2i 12	. 2 |- ( ( ph -> ps ) -> ( ph -> ( ps \/ ch ) ) )
3		olc 701	. . 3 |- ( ch -> ( ps \/ ch ) )
4	3	imim2i 12	. 2 |- ( ( ph -> ch ) -> ( ph -> ( ps \/ ch ) ) )
5	2, 4	jaoi 706	1 |- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) -> ( ph -> ( ps \/ ch ) ) )

Syntax hints:   -> wi 4   \/ wo 698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)