Theorem ralinexa

Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005)

	Ref	Expression
Assertion	ralinexa	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → ¬ 𝜓 ) ↔ ¬ ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		imnan	⊢ ( ( 𝜑 → ¬ 𝜓 ) ↔ ¬ ( 𝜑 ∧ 𝜓 ) )
2	1	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → ¬ 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ¬ ( 𝜑 ∧ 𝜓 ) )
3		ralnex	⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ ( 𝜑 ∧ 𝜓 ) ↔ ¬ ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → ¬ 𝜓 ) ↔ ¬ ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) )