Theorem orbidi 510

Description: Disjunction distributes over the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseer.lcs.mit.edu/lifschitz98calculational.html.

Assertion

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

Proof of Theorem orbidi

Step	Hyp	Ref	Expression
1		orc 225	. . . . 5 |- (ph -> (ph \/ ch))
2	1	a1d 14	. . . 4 |- (ph -> ((ph \/ ps) -> (ph \/ ch)))
3		orc 225	. . . . 5 |- (ph -> (ph \/ ps))
4	3	a1d 14	. . . 4 |- (ph -> ((ph \/ ch) -> (ph \/ ps)))
5	2, 4	impbid 397	. . 3 |- (ph -> ((ph \/ ps) <-> (ph \/ ch)))
6		id 9	. . . 4 |- ((ps <-> ch) -> (ps <-> ch))
7	6	orbi2d 466	. . 3 |- ((ps <-> ch) -> ((ph \/ ps) <-> (ph \/ ch)))
8	5, 7	jaoi 275	. 2 |- ((ph \/ (ps <-> ch)) -> ((ph \/ ps) <-> (ph \/ ch)))
9		pm2.85 439	. . . 4 |- (((ph \/ ps) -> (ph \/ ch)) -> (ph \/ (ps -> ch)))
10		pm2.85 439	. . . 4 |- (((ph \/ ch) -> (ph \/ ps)) -> (ph \/ (ch -> ps)))
11	9, 10	anim12i 268	. . 3 |- ((((ph \/ ps) -> (ph \/ ch)) /\ ((ph \/ ch) -> (ph \/ ps))) -> ((ph \/ (ps -> ch)) /\ (ph \/ (ch -> ps))))
12		bi 396	. . 3 |- (((ph \/ ps) <-> (ph \/ ch)) <-> (((ph \/ ps) -> (ph \/ ch)) /\ ((ph \/ ch) -> (ph \/ ps))))
13		bi 396	. . . . 5 |- ((ps <-> ch) <-> ((ps -> ch) /\ (ch -> ps)))
14	13	orbi2i 214	. . . 4 |- ((ph \/ (ps <-> ch)) <-> (ph \/ ((ps -> ch) /\ (ch -> ps))))
15		ordi 452	. . . 4 |- ((ph \/ ((ps -> ch) /\ (ch -> ps))) <-> ((ph \/ (ps -> ch)) /\ (ph \/ (ch -> ps))))
16	14, 15	bitr 151	. . 3 |- ((ph \/ (ps <-> ch)) <-> ((ph \/ (ps -> ch)) /\ (ph \/ (ch -> ps))))
17	11, 12, 16	3imtr4 192	. 2 |- (((ph \/ ps) <-> (ph \/ ch)) -> (ph \/ (ps <-> ch)))
18	8, 17	impbi 139	1 |- ((ph \/ (ps <-> ch)) <-> ((ph \/ ps) <-> (ph \/ ch)))

Syntax hints:   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196

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

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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