Theorem dfbi3 994

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.) (Proof shortened by NM, 29-Oct-2021.)

Assertion

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

Proof of Theorem dfbi3

Step	Hyp	Ref	Expression
1		con34b 306	. . 3 |- ( ( ps -> ph ) <-> ( -. ph -> -. ps ) )
2	1	anbi2i 730	. 2 |- ( ( ( ph -> ps ) /\ ( ps -> ph ) ) <-> ( ( ph -> ps ) /\ ( -. ph -> -. ps ) ) )
3		dfbi2 660	. 2 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
4		cases2 993	. 2 |- ( ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) <-> ( ( ph -> ps ) /\ ( -. ph -> -. ps ) ) )
5	2, 3, 4	3bitr4i 292	1 |- ( ( ph <-> ps ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by:  pm5.24  996  4exmidOLD  998  nanbi  1454  ifbi  4107  sqf11  24865  bj-dfbi4  32558  raaan2  41175  2reu4a  41189