Theorem nanim 1452

Description: Show equivalence between implication and the Nicod version. To derive nic-dfim 1594, apply nanbi 1454. (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 1451	. 2 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ↔ (𝜑 → (𝜓 ∧ 𝜓)))
2		anidmdbi 678	. 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜓)) ↔ (𝜑 → 𝜓))
3	1, 2	bitr2i 265	1 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ⊼ (𝜓 ⊼ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ⊼ 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-an 386  df-nan 1448

This theorem is referenced by:  nic-dfim  1594  nic-ax  1598  waj-ax  32413  lukshef-ax2  32414