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Theorem nic-dfim 1434
 Description: Define implication in terms of 'nand'. Analogous to ((φ ⊼ (ψ ⊼ ψ)) ↔ (φ → ψ)). 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 (((φ (ψ ψ)) (φψ)) (((φ (ψ ψ)) (φ (ψ ψ))) ((φψ) (φψ))))

Proof of Theorem nic-dfim
StepHypRef Expression
1 nanim 1292 . . 3 ((φψ) ↔ (φ (ψ ψ)))
21bicomi 193 . 2 ((φ (ψ ψ)) ↔ (φψ))
3 nanbi 1294 . 2 (((φ (ψ ψ)) ↔ (φψ)) ↔ (((φ (ψ ψ)) (φψ)) (((φ (ψ ψ)) (φ (ψ ψ))) ((φψ) (φψ)))))
42, 3mpbi 199 1 (((φ (ψ ψ)) (φψ)) (((φ (ψ ψ)) (φ (ψ ψ))) ((φψ) (φψ))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ⊼ wnan 1287 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288 This theorem is referenced by:  nic-stdmp  1455  nic-luk1  1456  nic-luk2  1457  nic-luk3  1458
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