Theorem pm2.85 582

Description: Theorem *2.85 of [WhiteheadRussell] p. 108.

Assertion

Ref	Expression
pm2.85	|- (((ph \/ ps) -> (ph \/ ch)) -> (ph \/ (ps -> ch)))

Proof of Theorem pm2.85

Step	Hyp	Ref	Expression
1		imor 234	. . 3 |- (((ph \/ ps) -> (ph \/ ch)) <-> (-. (ph \/ ps) \/ (ph \/ ch)))
2		pm2.48 280	. . . 4 |- (-. (ph \/ ps) -> (ph \/ -. ps))
3	2	orim1i 337	. . 3 |- ((-. (ph \/ ps) \/ (ph \/ ch)) -> ((ph \/ -. ps) \/ (ph \/ ch)))
4	1, 3	sylbi 199	. 2 |- (((ph \/ ps) -> (ph \/ ch)) -> ((ph \/ -. ps) \/ (ph \/ ch)))
5		imor 234	. . . 4 |- ((ps -> ch) <-> (-. ps \/ ch))
6	5	orbi2i 255	. . 3 |- ((ph \/ (ps -> ch)) <-> (ph \/ (-. ps \/ ch)))
7		orordi 266	. . 3 |- ((ph \/ (-. ps \/ ch)) <-> ((ph \/ -. ps) \/ (ph \/ ch)))
8	6, 7	bitr2 174	. 2 |- (((ph \/ -. ps) \/ (ph \/ ch)) <-> (ph \/ (ps -> ch)))
9	4, 8	sylib 198	1 |- (((ph \/ ps) -> (ph \/ ch)) -> (ph \/ (ps -> ch)))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222

This theorem is referenced by:  orbidi 747

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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