Theorem pm5.75 749

Description: Theorem *5.75 of [WhiteheadRussell] p. 126.

Assertion

Ref	Expression
pm5.75	|- (((ch -> -. ps) /\ (ph <-> (ps \/ ch))) -> ((ph /\ -. ps) <-> ch))

Proof of Theorem pm5.75

Step	Hyp	Ref	Expression
1		bi1 148	. . . 4 |- ((ph <-> (ps \/ ch)) -> (ph -> (ps \/ ch)))
2		pm5.6 688	. . . 4 |- (((ph /\ -. ps) -> ch) <-> (ph -> (ps \/ ch)))
3	1, 2	sylibr 200	. . 3 |- ((ph <-> (ps \/ ch)) -> ((ph /\ -. ps) -> ch))
4	3	adantl 388	. 2 |- (((ch -> -. ps) /\ (ph <-> (ps \/ ch))) -> ((ph /\ -. ps) -> ch))
5		bi2 149	. . . . . . . 8 |- ((ph <-> (ps \/ ch)) -> ((ps \/ ch) -> ph))
6		olc 268	. . . . . . . . 9 |- (ch -> (ps \/ ch))
7		olc 268	. . . . . . . . 9 |- (ph -> (ps \/ ph))
8	6, 7	imim12i 18	. . . . . . . 8 |- (((ps \/ ch) -> ph) -> (ch -> (ps \/ ph)))
9	5, 8	syl 10	. . . . . . 7 |- ((ph <-> (ps \/ ch)) -> (ch -> (ps \/ ph)))
10		pm5.6 688	. . . . . . 7 |- (((ch /\ -. ps) -> ph) <-> (ch -> (ps \/ ph)))
11	9, 10	sylibr 200	. . . . . 6 |- ((ph <-> (ps \/ ch)) -> ((ch /\ -. ps) -> ph))
12	11	exp3a 375	. . . . 5 |- ((ph <-> (ps \/ ch)) -> (ch -> (-. ps -> ph)))
13	12	a2d 13	. . . 4 |- ((ph <-> (ps \/ ch)) -> ((ch -> -. ps) -> (ch -> ph)))
14	13	impcom 351	. . 3 |- (((ch -> -. ps) /\ (ph <-> (ps \/ ch))) -> (ch -> ph))
15		pm3.26 319	. . 3 |- (((ch -> -. ps) /\ (ph <-> (ps \/ ch))) -> (ch -> -. ps))
16	14, 15	jcad 600	. 2 |- (((ch -> -. ps) /\ (ph <-> (ps \/ ch))) -> (ch -> (ph /\ -. ps)))
17	4, 16	impbid 516	1 |- (((ch -> -. ps) /\ (ph <-> (ps \/ ch))) -> ((ph /\ -. ps) <-> ch))

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

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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