Theorem alinexa

Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993)

	Ref	Expression
Assertion	alinexa	⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) ↔ ¬ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		imnang	⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) ↔ ∀ 𝑥 ¬ ( 𝜑 ∧ 𝜓 ) )
2		alnex	⊢ ( ∀ 𝑥 ¬ ( 𝜑 ∧ 𝜓 ) ↔ ¬ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) )
3	1 2	bitri	⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) ↔ ¬ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) )