Theorem ordi 908

Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 28-Nov-2013.)

Assertion

Ref	Expression
ordi	|- ( ( ph \/ ( ps /\ ch ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) )

Proof of Theorem ordi

Step	Hyp	Ref	Expression
1		jcab 907	. 2 |- ( ( -. ph -> ( ps /\ ch ) ) <-> ( ( -. ph -> ps ) /\ ( -. ph -> ch ) ) )
2		df-or 385	. 2 |- ( ( ph \/ ( ps /\ ch ) ) <-> ( -. ph -> ( ps /\ ch ) ) )
3		df-or 385	. . 3 |- ( ( ph \/ ps ) <-> ( -. ph -> ps ) )
4		df-or 385	. . 3 |- ( ( ph \/ ch ) <-> ( -. ph -> ch ) )
5	3, 4	anbi12i 733	. 2 |- ( ( ( ph \/ ps ) /\ ( ph \/ ch ) ) <-> ( ( -. ph -> ps ) /\ ( -. ph -> ch ) ) )
6	1, 2, 5	3bitr4i 292	1 |- ( ( ph \/ ( ps /\ ch ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ 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-or 385  df-an 386

This theorem is referenced by:  ordir  909  orddi  913  pm5.63  959  pm4.43  968  cadan  1548  undi  3874  undif3  3888  undif4  4035  elnn1uz2  11765  or3di  29307  ifpan23  37804  ifpidg  37836  ifpim123g  37845