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Mirrors > Home > MPE Home > Th. List > rmoeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2139, ax-11 2155, and ax-12 2175. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2108. (Revised by Wolf Lammen, 12-Mar-2025.) |
Ref | Expression |
---|---|
rmoeq1 | ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2728 | . . . . . 6 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
2 | 1 | biimpi 216 | . . . . 5 ⊢ (𝐴 = 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
3 | anbi1 633 | . . . . . . 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 1808 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → ∀𝑥(((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝑧))) |
6 | albi 1815 | . . . . 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 1919 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑧∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝑧))) |
9 | df-mo 2538 | . . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑧∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑧)) | |
10 | df-mo 2538 | . . 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 3378 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
13 | df-rmo 3378 | . 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
14 | 11, 12, 13 | 3bitr4g 314 | 1 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∃*wmo 2536 ∃*wrmo 3377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-mo 2538 df-cleq 2727 df-rmo 3378 |
This theorem is referenced by: reueq1 3415 rmoeqd 3418 rmosn 4724 rmoeqdv 36194 poimirlem2 37609 |
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