Theorem expandrex

Description: Expand a restricted existential quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023)

	Ref	Expression
Hypothesis	expandrex.1	⊢ ( 𝜑 ↔ 𝜓 )
Assertion	expandrex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		expandrex.1	⊢ ( 𝜑 ↔ 𝜓 )
2		notnotb	⊢ ( 𝜓 ↔ ¬ ¬ 𝜓 )
3	1 2	bitri	⊢ ( 𝜑 ↔ ¬ ¬ 𝜓 )
4	3	expandrexn	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ¬ 𝜓 ) )