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)) ⊼ (φ → ψ)) ⊼ (((φ ⊼ (ψ ⊼ ψ)) ⊼ (φ ⊼ (ψ ⊼ ψ))) ⊼ ((φ → ψ) ⊼ (φ → ψ))))

Proof of Theorem nic-dfim

Step	Hyp	Ref	Expression
1		nanim 1292	. . 3 ⊢ ((φ → ψ) ↔ (φ ⊼ (ψ ⊼ ψ)))
2	1	bicomi 193	. 2 ⊢ ((φ ⊼ (ψ ⊼ ψ)) ↔ (φ → ψ))
3		nanbi 1294	. 2 ⊢ (((φ ⊼ (ψ ⊼ ψ)) ↔ (φ → ψ)) ↔ (((φ ⊼ (ψ ⊼ ψ)) ⊼ (φ → ψ)) ⊼ (((φ ⊼ (ψ ⊼ ψ)) ⊼ (φ ⊼ (ψ ⊼ ψ))) ⊼ ((φ → ψ) ⊼ (φ → ψ)))))
4	2, 3	mpbi 199	1 ⊢ (((φ ⊼ (ψ ⊼ ψ)) ⊼ (φ → ψ)) ⊼ (((φ ⊼ (ψ ⊼ ψ)) ⊼ (φ ⊼ (ψ ⊼ ψ))) ⊼ ((φ → ψ) ⊼ (φ → ψ))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ⊼ wnan 1287

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288

This theorem is used by:  nic-stdmp  1455  nic-luk1  1456  nic-luk2  1457  nic-luk3  1458