Theorem nannan 1487

Description: Nested alternative denials. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 26-Jun-2020.)

Assertion

Ref	Expression
nannan	⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒)))

Proof of Theorem nannan

Step	Hyp	Ref	Expression
1		nanimn 1484	. 2 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ⊼ 𝜒)))
2		nanan 1483	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ ¬ (𝜓 ⊼ 𝜒))
3	2	imbi2i 338	. 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ⊼ 𝜒)))
4	1, 3	bitr4i 280	1 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ⊼ wnan 1481

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 209  df-an 399  df-nan 1482

This theorem is referenced by:  nanim  1488  nanbi  1490  nanass  1500  nic-mp  1672  nic-ax  1674  waj-ax  33764  lukshef-ax2  33765  arg-ax  33766