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 → ( ∃* 𝑥𝐴 𝜑 ↔ ∃* 𝑥𝐵 𝜑 ) )

Proof

Step Hyp Ref Expression
1 dfcleq ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ( 𝑥𝐴𝑥𝐵 ) )
2 1 biimpi ( 𝐴 = 𝐵 → ∀ 𝑥 ( 𝑥𝐴𝑥𝐵 ) )
3 anbi1 ( ( 𝑥𝐴𝑥𝐵 ) → ( ( 𝑥𝐴𝜑 ) ↔ ( 𝑥𝐵𝜑 ) ) )
4 3 imbi1d ( ( 𝑥𝐴𝑥𝐵 ) → ( ( ( 𝑥𝐴𝜑 ) → 𝑥 = 𝑧 ) ↔ ( ( 𝑥𝐵𝜑 ) → 𝑥 = 𝑧 ) ) )
5 4 alimi ( ∀ 𝑥 ( 𝑥𝐴𝑥𝐵 ) → ∀ 𝑥 ( ( ( 𝑥𝐴𝜑 ) → 𝑥 = 𝑧 ) ↔ ( ( 𝑥𝐵𝜑 ) → 𝑥 = 𝑧 ) ) )
6 albi ( ∀ 𝑥 ( ( ( 𝑥𝐴𝜑 ) → 𝑥 = 𝑧 ) ↔ ( ( 𝑥𝐵𝜑 ) → 𝑥 = 𝑧 ) ) → ( ∀ 𝑥 ( ( 𝑥𝐴𝜑 ) → 𝑥 = 𝑧 ) ↔ ∀ 𝑥 ( ( 𝑥𝐵𝜑 ) → 𝑥 = 𝑧 ) ) )
7 2 5 6 3syl ( 𝐴 = 𝐵 → ( ∀ 𝑥 ( ( 𝑥𝐴𝜑 ) → 𝑥 = 𝑧 ) ↔ ∀ 𝑥 ( ( 𝑥𝐵𝜑 ) → 𝑥 = 𝑧 ) ) )
8 7 exbidv ( 𝐴 = 𝐵 → ( ∃ 𝑧𝑥 ( ( 𝑥𝐴𝜑 ) → 𝑥 = 𝑧 ) ↔ ∃ 𝑧𝑥 ( ( 𝑥𝐵𝜑 ) → 𝑥 = 𝑧 ) ) )
9 df-mo ( ∃* 𝑥 ( 𝑥𝐴𝜑 ) ↔ ∃ 𝑧𝑥 ( ( 𝑥𝐴𝜑 ) → 𝑥 = 𝑧 ) )
10 df-mo ( ∃* 𝑥 ( 𝑥𝐵𝜑 ) ↔ ∃ 𝑧𝑥 ( ( 𝑥𝐵𝜑 ) → 𝑥 = 𝑧 ) )
11 8 9 10 3bitr4g ( 𝐴 = 𝐵 → ( ∃* 𝑥 ( 𝑥𝐴𝜑 ) ↔ ∃* 𝑥 ( 𝑥𝐵𝜑 ) ) )
12 df-rmo ( ∃* 𝑥𝐴 𝜑 ↔ ∃* 𝑥 ( 𝑥𝐴𝜑 ) )
13 df-rmo ( ∃* 𝑥𝐵 𝜑 ↔ ∃* 𝑥 ( 𝑥𝐵𝜑 ) )
14 11 12 13 3bitr4g ( 𝐴 = 𝐵 → ( ∃* 𝑥𝐴 𝜑 ↔ ∃* 𝑥𝐵 𝜑 ) )