Theorem pm5.17 666

Description: Theorem *5.17 of [WhiteheadRussell] p. 124.

Assertion

Ref	Expression
pm5.17	|- (((ph \/ ps) /\ -. (ph /\ ps)) <-> (ph <-> -. ps))

Proof of Theorem pm5.17

Step	Hyp	Ref	Expression
1		orcom 246	. . . 4 |- ((ph \/ ps) <-> (ps \/ ph))
2		df-or 224	. . . 4 |- ((ps \/ ph) <-> (-. ps -> ph))
3	1, 2	bitr 173	. . 3 |- ((ph \/ ps) <-> (-. ps -> ph))
4		imnan 242	. . . 4 |- ((ph -> -. ps) <-> -. (ph /\ ps))
5	4	bicomi 172	. . 3 |- (-. (ph /\ ps) <-> (ph -> -. ps))
6	3, 5	anbi12i 481	. 2 |- (((ph \/ ps) /\ -. (ph /\ ps)) <-> ((-. ps -> ph) /\ (ph -> -. ps)))
7		bi 513	. 2 |- ((-. ps <-> ph) <-> ((-. ps -> ph) /\ (ph -> -. ps)))
8		bicom 518	. 2 |- ((-. ps <-> ph) <-> (ph <-> -. ps))
9	6, 7, 8	3bitr2 179	1 |- (((ph \/ ps) /\ -. (ph /\ ps)) <-> (ph <-> -. ps))

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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