Theorem nannan 1451

Description: Lemma for handling nested 'nand's. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 7-Mar-2020.)

Assertion

Ref	Expression
nannan	|- ( ( ph -/\ ( ch -/\ ps ) ) <-> ( ph -> ( ch /\ ps ) ) )

Proof of Theorem nannan

Step	Hyp	Ref	Expression
1		imnan 438	. 2 |- ( ( ph -> -. ( ch -/\ ps ) ) <-> -. ( ph /\ ( ch -/\ ps ) ) )
2		nanan 1449	. . 3 |- ( ( ch /\ ps ) <-> -. ( ch -/\ ps ) )
3	2	imbi2i 326	. 2 |- ( ( ph -> ( ch /\ ps ) ) <-> ( ph -> -. ( ch -/\ ps ) ) )
4		df-nan 1448	. 2 |- ( ( ph -/\ ( ch -/\ ps ) ) <-> -. ( ph /\ ( ch -/\ ps ) ) )
5	1, 3, 4	3bitr4ri 293	1 |- ( ( ph -/\ ( ch -/\ ps ) ) <-> ( ph -> ( ch /\ 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:  nanim  1452  nanbi  1454  nic-mp  1596  nic-ax  1598  waj-ax  32413  lukshef-ax2  32414  arg-ax  32415  rp-fakenanass  37860