Theorem nanim 1292

Description: Show equivalence between implication and the Nicod version. To derive nic-dfim 1434, apply nanbi 1294. (Contributed by Jeff Hoffman, 19-Nov-2007.)

Assertion

Ref	Expression
nanim	⊢ ((φ → ψ) ↔ (φ ⊼ (ψ ⊼ ψ)))

Proof of Theorem nanim

Step	Hyp	Ref	Expression
1		nannan 1291	. 2 ⊢ ((φ ⊼ (ψ ⊼ ψ)) ↔ (φ → (ψ ∧ ψ)))
2		anidmdbi 627	. 2 ⊢ ((φ → (ψ ∧ ψ)) ↔ (φ → ψ))
3	1, 2	bitr2i 241	1 ⊢ ((φ → ψ) ↔ (φ ⊼ (ψ ⊼ ψ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ⊼ wnan 1287

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 177  df-an 360  df-nan 1288

This theorem is referenced by:  nic-dfim  1434  nic-ax  1438