Theorem ordi 834

Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-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 833	. 2 |- ((-. ph -> (ps /\ ch)) <-> ((-. ph -> ps) /\ (-. ph -> ch)))
2		df-or 359	. 2 |- ((ph \/ (ps /\ ch)) <-> (-. ph -> (ps /\ ch)))
3		df-or 359	. . 3 |- ((ph \/ ps) <-> (-. ph -> ps))
4		df-or 359	. . 3 |- ((ph \/ ch) <-> (-. ph -> ch))
5	3, 4	anbi12i 678	. 2 |- (((ph \/ ps) /\ (ph \/ ch)) <-> ((-. ph -> ps) /\ (-. ph -> ch)))
6	1, 2, 5	3bitr4i 268	1 |- ((ph \/ (ps /\ ch)) <-> ((ph \/ ps) /\ (ph \/ ch)))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358

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 177  df-or 359  df-an 360

This theorem is referenced by:  ordir  835  orddi  839  pm5.63  890  pm4.43  893  cadan  1392  undi  3503  undif4  3608