Theorem rmoeq1 3425

Description: Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2141, ax-11 2158, and ax-12 2178. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2110. (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 2733	. . . . . 6 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	biimpi 216	. . . . 5 ⊢ (𝐴 = 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3		anbi1 632	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)))
4	3	imbi1d 341	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝑧)))
5	4	alimi 1809	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → ∀𝑥(((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝑧)))
6		albi 1816	. . . . 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 18	. . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧) ↔ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝑧)))
8	7	exbidv 1920	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑧∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝑧)))
9		df-mo 2543	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑧∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧))
10		df-mo 2543	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) ↔ ∃𝑧∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝑧))
11	8, 9, 10	3bitr4g 314	. 2 ⊢ (𝐴 = 𝐵 → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
12		df-rmo 3388	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
13		df-rmo 3388	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
14	11, 12, 13	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃*wmo 2541  ∃*wrmo 3387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-mo 2543  df-cleq 2732  df-rmo 3388

This theorem is referenced by:  reueq1  3426  rmoeqd  3429  rmosn  4744  rmoeqdv  36176  poimirlem2  37582