Description: This theorem "defines" implication in terms of 'nand'. Analogous to nanim . 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.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | nic-dfim | |- ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -> ps ) ) -/\ ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -/\ ( ps -/\ ps ) ) ) -/\ ( ( ph -> ps ) -/\ ( ph -> ps ) ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nanim | |- ( ( ph -> ps ) <-> ( ph -/\ ( ps -/\ ps ) ) ) |
|
| 2 | 1 | bicomi | |- ( ( ph -/\ ( ps -/\ ps ) ) <-> ( ph -> ps ) ) |
| 3 | nanbi | |- ( ( ( ph -/\ ( ps -/\ ps ) ) <-> ( ph -> ps ) ) <-> ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -> ps ) ) -/\ ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -/\ ( ps -/\ ps ) ) ) -/\ ( ( ph -> ps ) -/\ ( ph -> ps ) ) ) ) ) |
|
| 4 | 2 3 | mpbi | |- ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -> ps ) ) -/\ ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -/\ ( ps -/\ ps ) ) ) -/\ ( ( ph -> ps ) -/\ ( ph -> ps ) ) ) ) |