Theorem mobi

Description: Equivalence theorem for the at-most-one quantifier. (Contributed by BJ, 7-Oct-2022) (Proof shortened by Wolf Lammen, 18-Feb-2023)

	Ref	Expression
Assertion	mobi	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( ∃* 𝑥 𝜑 ↔ ∃* 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		albiim	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑥 ( 𝜓 → 𝜑 ) ) )
2		moim	⊢ ( ∀ 𝑥 ( 𝜓 → 𝜑 ) → ( ∃* 𝑥 𝜑 → ∃* 𝑥 𝜓 ) )
3		moim	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃* 𝑥 𝜓 → ∃* 𝑥 𝜑 ) )
4	2 3	impbid21d	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ( 𝜓 → 𝜑 ) → ( ∃* 𝑥 𝜑 ↔ ∃* 𝑥 𝜓 ) ) )
5	4	imp	⊢ ( ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑥 ( 𝜓 → 𝜑 ) ) → ( ∃* 𝑥 𝜑 ↔ ∃* 𝑥 𝜓 ) )
6	1 5	sylbi	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( ∃* 𝑥 𝜑 ↔ ∃* 𝑥 𝜓 ) )