Theorem expandexn

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

	Ref	Expression
Hypothesis	expandexn.1	⊢ ( 𝜑 ↔ ¬ 𝜓 )
Assertion	expandexn	⊢ ( ∃ 𝑥 𝜑 ↔ ¬ ∀ 𝑥 𝜓 )

Proof

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