Theorem expandrexn

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

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

Proof

Step	Hyp	Ref	Expression
1		expandrexn.1	⊢ ( 𝜑 ↔ ¬ 𝜓 )
2	1	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 ¬ 𝜓 )
3		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ¬ 𝜓 ) )
4		exanali	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ¬ 𝜓 ) ↔ ¬ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
5	2 3 4	3bitri	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )