Theorem rmoeq1

Description: Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023) Avoid ax-8 . (Revised by Wolf Lammen, 12-Mar-2025)

	Ref	Expression
Assertion	rmoeq1	⊢ ( 𝐴 = 𝐵 → ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑥 ∈ 𝐵 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		dfcleq	⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
2	1	biimpi	⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
3		anbi1	⊢ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) )
4	3	imbi1d	⊢ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ) )
5	4	alimi	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) → ∀ 𝑥 ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ) )
6		albi	⊢ ( ∀ 𝑥 ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ) → ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ) )
7	2 5 6	3syl	⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ) )
8	7	exbidv	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑧 ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ↔ ∃ 𝑧 ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝑥 = 𝑧 ) ) )
9		dfmo	⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃ 𝑧 ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) )
10		dfmo	⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ↔ ∃ 𝑧 ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝑥 = 𝑧 ) )
11	8 9 10	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) )
12		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
13		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐵 𝜑 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
14	11 12 13	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑥 ∈ 𝐵 𝜑 ) )