Theorem nannan 1504

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		dfnan2 1501	. 2 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ⊼ 𝜒)))
2		nanan 1500	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ ¬ (𝜓 ⊼ 𝜒))
3	2	imbi2i 337	. 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ⊼ 𝜒)))
4	1, 3	bitr4i 279	1 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ⊼ wnan 1498

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 208  df-an 397  df-nan 1499

This theorem is referenced by:  nanim  1505  nanbi  1507  nanass  1517  nic-mp  1678  nic-ax  1680  waj-ax  36649  lukshef-ax2  36650  arg-ax  36651