Theorem nannan 1619

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 1615	. 2 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ⊼ 𝜒)))
2		nanan 1614	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ ¬ (𝜓 ⊼ 𝜒))
3	2	imbi2i 328	. 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ⊼ 𝜒)))
4	1, 3	bitr4i 270	1 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ⊼ wnan 1612

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 199  df-an 387  df-nan 1613

This theorem is referenced by:  nanim  1621  nanbi  1624  nanass  1635  nanassOLD  1636  nic-mp  1770  nic-ax  1772  waj-ax  32935  lukshef-ax2  32936  arg-ax  32937