Theorem nic-dfim 1594

Description: Define implication in terms of 'nand'. Analogous to ( ( ph -/\ ( ps -/\ ps ) ) <-> ( ph -> ps ) ). In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition ($a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nic-dfim	|- ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -> ps ) ) -/\ ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -/\ ( ps -/\ ps ) ) ) -/\ ( ( ph -> ps ) -/\ ( ph -> ps ) ) ) )

Proof of Theorem nic-dfim

Step	Hyp	Ref	Expression
1		nanim 1452	. . 3 |- ( ( ph -> ps ) <-> ( ph -/\ ( ps -/\ ps ) ) )
2	1	bicomi 214	. 2 |- ( ( ph -/\ ( ps -/\ ps ) ) <-> ( ph -> ps ) )
3		nanbi 1454	. 2 |- ( ( ( ph -/\ ( ps -/\ ps ) ) <-> ( ph -> ps ) ) <-> ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -> ps ) ) -/\ ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -/\ ( ps -/\ ps ) ) ) -/\ ( ( ph -> ps ) -/\ ( ph -> ps ) ) ) ) )
4	2, 3	mpbi 220	1 |- ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -> ps ) ) -/\ ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -/\ ( ps -/\ ps ) ) ) -/\ ( ( ph -> ps ) -/\ ( ph -> ps ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   -/\ 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-or 385  df-an 386  df-nan 1448

This theorem is referenced by:  nic-stdmp  1615  nic-luk1  1616  nic-luk2  1617  nic-luk3  1618