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Mirrors > Home > MPE Home > Th. List > reurab | Structured version Visualization version GIF version |
Description: Restricted existential uniqueness of a restricted abstraction. (Contributed by Scott Fenton, 8-Aug-2024.) |
Ref | Expression |
---|---|
reurab.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
reurab | ⊢ (∃!𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓}𝜒 ↔ ∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reurab.1 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | bicomd 223 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑)) |
3 | 2 | equcoms 2017 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑)) |
4 | 3 | elrab 3695 | . . . . 5 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
5 | 4 | anbi1i 624 | . . . 4 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ∧ 𝜒) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜒)) |
6 | anass 468 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒))) | |
7 | 5, 6 | bitri 275 | . . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒))) |
8 | 7 | eubii 2583 | . 2 ⊢ (∃!𝑥(𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ∧ 𝜒) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒))) |
9 | df-reu 3379 | . 2 ⊢ (∃!𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓}𝜒 ↔ ∃!𝑥(𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ∧ 𝜒)) | |
10 | df-reu 3379 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜒) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒))) | |
11 | 8, 9, 10 | 3bitr4i 303 | 1 ⊢ (∃!𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓}𝜒 ↔ ∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ∃!weu 2566 ∃!wreu 3376 {crab 3433 |
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-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-reu 3379 df-rab 3434 df-v 3480 |
This theorem is referenced by: eqscut 27865 |
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