Theorem nanim 1498

Description: Implication in terms of alternative denial. (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 1497	. 2 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ↔ (𝜑 → (𝜓 ∧ 𝜓)))
2		anidmdbi 565	. 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜓)) ↔ (𝜑 → 𝜓))
3	1, 2	bitr2i 276	1 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ⊼ (𝜓 ⊼ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ⊼ wnan 1491

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 207  df-an 396  df-nan 1492

This theorem is referenced by:  nic-dfim  1669  nic-ax  1673  waj-ax  36415  lukshef-ax2  36416