Theorem nannanOLD 1617

Description: Obsolete proof of nannan 1616 as of 19-Oct-2022. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 7-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem nannanOLD

Step	Hyp	Ref	Expression
1		imnan 389	. 2 ⊢ ((𝜑 → ¬ (𝜓 ⊼ 𝜒)) ↔ ¬ (𝜑 ∧ (𝜓 ⊼ 𝜒)))
2		nanan 1611	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ ¬ (𝜓 ⊼ 𝜒))
3	2	imbi2i 328	. 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ⊼ 𝜒)))
4		df-nan 1610	. 2 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ ¬ (𝜑 ∧ (𝜓 ⊼ 𝜒)))
5	1, 3, 4	3bitr4ri 296	1 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 385   ⊼ wnan 1609

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 386  df-nan 1610

This theorem is referenced by: (None)