Theorem pm5.17 858

Description: Theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Jan-2013.)

Assertion

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

Proof of Theorem pm5.17

Step	Hyp	Ref	Expression
1		bicom 191	. 2 |- ((ph <-> -. ps) <-> (-. ps <-> ph))
2		dfbi2 609	. 2 |- ((-. ps <-> ph) <-> ((-. ps -> ph) /\ (ph -> -. ps)))
3		orcom 376	. . . 4 |- ((ph \/ ps) <-> (ps \/ ph))
4		df-or 359	. . . 4 |- ((ps \/ ph) <-> (-. ps -> ph))
5	3, 4	bitr2i 241	. . 3 |- ((-. ps -> ph) <-> (ph \/ ps))
6		imnan 411	. . 3 |- ((ph -> -. ps) <-> -. (ph /\ ps))
7	5, 6	anbi12i 678	. 2 |- (((-. ps -> ph) /\ (ph -> -. ps)) <-> ((ph \/ ps) /\ -. (ph /\ ps)))
8	1, 2, 7	3bitrri 263	1 |- (((ph \/ ps) /\ -. (ph /\ ps)) <-> (ph <-> -. ps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is referenced by:  nbi2  862