Theorem ordi 594

Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119.

Assertion

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

Proof of Theorem ordi

Step	Hyp	Ref	Expression
1		pm3.26 319	. . . 4 |- ((ps /\ ch) -> ps)
2	1	orim2i 338	. . 3 |- ((ph \/ (ps /\ ch)) -> (ph \/ ps))
3		pm3.27 323	. . . 4 |- ((ps /\ ch) -> ch)
4	3	orim2i 338	. . 3 |- ((ph \/ (ps /\ ch)) -> (ph \/ ch))
5	2, 4	jca 288	. 2 |- ((ph \/ (ps /\ ch)) -> ((ph \/ ps) /\ (ph \/ ch)))
6		df-or 224	. . . 4 |- ((ph \/ ps) <-> (-. ph -> ps))
7		pm3.43i 287	. . . . 5 |- ((-. ph -> ps) -> ((-. ph -> ch) -> (-. ph -> (ps /\ ch))))
8		df-or 224	. . . . 5 |- ((ph \/ ch) <-> (-. ph -> ch))
9		df-or 224	. . . . 5 |- ((ph \/ (ps /\ ch)) <-> (-. ph -> (ps /\ ch)))
10	7, 8, 9	3imtr4g 551	. . . 4 |- ((-. ph -> ps) -> ((ph \/ ch) -> (ph \/ (ps /\ ch))))
11	6, 10	sylbi 199	. . 3 |- ((ph \/ ps) -> ((ph \/ ch) -> (ph \/ (ps /\ ch))))
12	11	imp 350	. 2 |- (((ph \/ ps) /\ (ph \/ ch)) -> (ph \/ (ps /\ ch)))
13	5, 12	impbi 157	1 |- ((ph \/ (ps /\ ch)) <-> ((ph \/ ps) /\ (ph \/ ch)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223

This theorem is referenced by:  ordir 595  jcab 596  andi 602  orddi 604  orbidi 741  undi 2242  undif4 2315

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225

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