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Metamath Proof Explorer

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 df-mo ⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃ 𝑧 ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 = 𝑧 ) )
10 df-mo ⊢ ( ∃* 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 ) )