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Mirrors > Home > MPE Home > Th. List > reuhypd | Structured version Visualization version GIF version |
Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 7399. (Contributed by NM, 16-Jan-2012.) |
Ref | Expression |
---|---|
reuhypd.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶) |
reuhypd.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) |
Ref | Expression |
---|---|
reuhypd | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reuhypd.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶) | |
2 | 1 | elexd 3494 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ V) |
3 | eueq 3704 | . . . 4 ⊢ (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵) | |
4 | 2, 3 | sylib 217 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 𝑦 = 𝐵) |
5 | eleq1 2821 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
6 | 1, 5 | syl5ibrcom 246 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑦 = 𝐵 → 𝑦 ∈ 𝐶)) |
7 | 6 | pm4.71rd 563 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑦 = 𝐵 ↔ (𝑦 ∈ 𝐶 ∧ 𝑦 = 𝐵))) |
8 | reuhypd.2 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) | |
9 | 8 | 3expa 1118 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) |
10 | 9 | pm5.32da 579 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴) ↔ (𝑦 ∈ 𝐶 ∧ 𝑦 = 𝐵))) |
11 | 7, 10 | bitr4d 281 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑦 = 𝐵 ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴))) |
12 | 11 | eubidv 2580 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (∃!𝑦 𝑦 = 𝐵 ↔ ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴))) |
13 | 4, 12 | mpbid 231 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) |
14 | df-reu 3377 | . 2 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = 𝐴 ↔ ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) | |
15 | 13, 14 | sylibr 233 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃!weu 2562 ∃!wreu 3374 Vcvv 3474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1089 df-tru 1544 df-ex 1782 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-reu 3377 df-v 3476 |
This theorem is referenced by: reuhyp 5418 riotaocN 38074 |
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