Theorem rexanali

Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005) (Proof shortened by Wolf Lammen, 27-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		dfrex2	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ ¬ 𝜓 ) ↔ ¬ ∀ 𝑥 ∈ 𝐴 ¬ ( 𝜑 ∧ ¬ 𝜓 ) )
2		iman	⊢ ( ( 𝜑 → 𝜓 ) ↔ ¬ ( 𝜑 ∧ ¬ 𝜓 ) )
3	2	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ¬ ( 𝜑 ∧ ¬ 𝜓 ) )
4	1 3	xchbinxr	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ ¬ 𝜓 ) ↔ ¬ ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) )