Theorem nannan

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

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

Proof

Step	Hyp	Ref	Expression
1		nanimn	⊢ ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜒 ) ) ↔ ( 𝜑 → ¬ ( 𝜓 ⊼ 𝜒 ) ) )
2		nanan	⊢ ( ( 𝜓 ∧ 𝜒 ) ↔ ¬ ( 𝜓 ⊼ 𝜒 ) )
3	2	imbi2i	⊢ ( ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) ↔ ( 𝜑 → ¬ ( 𝜓 ⊼ 𝜒 ) ) )
4	1 3	bitr4i	⊢ ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜒 ) ) ↔ ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) )