Theorem rmoeq1 3407

Description: Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2182, ax-11 2198, and ax-12 2219. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2151. (Revised by Wolf Lammen, 12-Mar-2025.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rmoeq1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2762	. . . . . 6 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	biimpi 219	. . . . 5 ⊢ (𝐴 = 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3		anbi1 644	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)))
4	3	imbi1d 344	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝑧)))
5	4	alimi 1838	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → ∀𝑥(((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝑧)))
6		albi 1845	. . . . 5 ⊢ (∀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 19	. . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧) ↔ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝑧)))
8	7	exbidv 1948	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑧∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝑧)))
9		dfmo 2574	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑧∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧))
10		dfmo 2574	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) ↔ ∃𝑧∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝑧))
11	8, 9, 10	3bitr4g 317	. 2 ⊢ (𝐴 = 𝐵 → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
12		df-rmo 3376	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
13		df-rmo 3376	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
14	11, 12, 13	3bitr4g 317	1 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃*wmo 2571  ∃*wrmo 3375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-mo 2573  df-cleq 2761  df-rmo 3376

This theorem is referenced by:  reueq1  3408  rmoeqd  3409  rmosn  4690  rmoeqdv  36612  poimirlem2  38160