Theorem rmoeq1 3397

Description: Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2178, ax-11 2194, and ax-12 2213. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2147. (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 2754	. . . . . 6 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	biimpi 219	. . . . 5 ⊢ (𝐴 = 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3		anbi1 645	. . . . . . 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 1844	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → ∀𝑥(((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝑧)))
6		albi 1851	. . . . 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 1954	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑧∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝑧)))
9		dfmo 2566	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑧∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧))
10		dfmo 2566	. . 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 3366	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
13		df-rmo 3366	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
14	11, 12, 13	3bitr4g 317	1 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2563  ∃*wrmo 3365

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2565  df-cleq 2753  df-rmo 3366

This theorem is used by:  reueq1  3398  rmoeqd  3399  rmosn  4680  rmoeqdv  37001  poimirlem2  38540