Theorem dfbi3 863

Description: An alternate definition of the biconditional. Theorem *5.23 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)

Assertion

Ref	Expression
dfbi3	|- ((ph <-> ps) <-> ((ph /\ ps) \/ (-. ph /\ -. ps)))

Proof of Theorem dfbi3

Step	Hyp	Ref	Expression
1		xor 861	. 2 |- (-. (ph <-> -. ps) <-> ((ph /\ -. -. ps) \/ (-. ps /\ -. ph)))
2		pm5.18 345	. 2 |- ((ph <-> ps) <-> -. (ph <-> -. ps))
3		notnot 282	. . . 4 |- (ps <-> -. -. ps)
4	3	anbi2i 675	. . 3 |- ((ph /\ ps) <-> (ph /\ -. -. ps))
5		ancom 437	. . 3 |- ((-. ph /\ -. ps) <-> (-. ps /\ -. ph))
6	4, 5	orbi12i 507	. 2 |- (((ph /\ ps) \/ (-. ph /\ -. ps)) <-> ((ph /\ -. -. ps) \/ (-. ps /\ -. ph)))
7	1, 2, 6	3bitr4i 268	1 |- ((ph <-> ps) <-> ((ph /\ ps) \/ (-. ph /\ -. ps)))

Syntax hints:  -. wn 3   <-> 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:  pm5.24  864  4exmid  905  nanbi  1294  ifbi  3680