Theorem ifpdfnan 37831

Description: Define nand as conditional logic operator. (Contributed by RP, 20-Apr-2020.)

Assertion

Ref	Expression
ifpdfnan	|- ( ( ph -/\ ps ) <-> if- ( ph , -. ps , T. ) )

Proof of Theorem ifpdfnan

Step	Hyp	Ref	Expression
1		df-nan 1448	. 2 |- ( ( ph -/\ ps ) <-> -. ( ph /\ ps ) )
2		ifpdfan 37810	. . 3 |- ( ( ph /\ ps ) <-> if- ( ph , ps , F. ) )
3	2	notbii 310	. 2 |- ( -. ( ph /\ ps ) <-> -. if- ( ph , ps , F. ) )
4		ifpnot23 37823	. . 3 |- ( -. if- ( ph , ps , F. ) <-> if- ( ph , -. ps , -. F. ) )
5		notfal 1519	. . . 4 |- ( -. F. <-> T. )
6		ifpbi3 37812	. . . 4 |- ( ( -. F. <-> T. ) -> (if- ( ph , -. ps , -. F. ) <-> if- ( ph , -. ps , T. ) ) )
7	5, 6	ax-mp 5	. . 3 |- (if- ( ph , -. ps , -. F. ) <-> if- ( ph , -. ps , T. ) )
8	4, 7	bitri 264	. 2 |- ( -. if- ( ph , ps , F. ) <-> if- ( ph , -. ps , T. ) )
9	1, 3, 8	3bitri 286	1 |- ( ( ph -/\ ps ) <-> if- ( ph , -. ps , T. ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384  if-wif 1012   -/\ wnan 1447   T. wtru 1484   F. wfal 1488

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  df-ifp 1013  df-nan 1448  df-tru 1486  df-fal 1489

This theorem is referenced by: (None)