Theorem exanali

Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996) (Proof shortened by Wolf Lammen, 4-Sep-2014)

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

Proof

Step	Hyp	Ref	Expression
1		annim	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) ↔ ¬ ( 𝜑 → 𝜓 ) )
2	1	exbii	⊢ ( ∃ 𝑥 ( 𝜑 ∧ ¬ 𝜓 ) ↔ ∃ 𝑥 ¬ ( 𝜑 → 𝜓 ) )
3		exnal	⊢ ( ∃ 𝑥 ¬ ( 𝜑 → 𝜓 ) ↔ ¬ ∀ 𝑥 ( 𝜑 → 𝜓 ) )
4	2 3	bitri	⊢ ( ∃ 𝑥 ( 𝜑 ∧ ¬ 𝜓 ) ↔ ¬ ∀ 𝑥 ( 𝜑 → 𝜓 ) )