Theorem imnand2 32399

Description: An -> nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)

Assertion

Ref	Expression
imnand2	|- ( ( -. ph -> ps ) <-> ( ( ph -/\ ph ) -/\ ( ps -/\ ps ) ) )

Proof of Theorem imnand2

Step	Hyp	Ref	Expression
1		nannot 1453	. . . 4 |- ( -. ph <-> ( ph -/\ ph ) )
2		nannot 1453	. . . 4 |- ( -. ps <-> ( ps -/\ ps ) )
3	1, 2	anbi12i 733	. . 3 |- ( ( -. ph /\ -. ps ) <-> ( ( ph -/\ ph ) /\ ( ps -/\ ps ) ) )
4	3	notbii 310	. 2 |- ( -. ( -. ph /\ -. ps ) <-> -. ( ( ph -/\ ph ) /\ ( ps -/\ ps ) ) )
5		iman 440	. 2 |- ( ( -. ph -> ps ) <-> -. ( -. ph /\ -. ps ) )
6		df-nan 1448	. 2 |- ( ( ( ph -/\ ph ) -/\ ( ps -/\ ps ) ) <-> -. ( ( ph -/\ ph ) /\ ( ps -/\ ps ) ) )
7	4, 5, 6	3bitr4i 292	1 |- ( ( -. ph -> ps ) <-> ( ( ph -/\ ph ) -/\ ( ps -/\ ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   -/\ wnan 1447

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-an 386  df-nan 1448

This theorem is referenced by: (None)