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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 6784. (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 3366 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ V) |
3 | eueq 3530 | . . . 4 ⊢ (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵) | |
4 | 2, 3 | sylib 208 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 𝑦 = 𝐵) |
5 | eleq1 2838 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
6 | 1, 5 | syl5ibrcom 237 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑦 = 𝐵 → 𝑦 ∈ 𝐶)) |
7 | 6 | pm4.71rd 552 | . . . . 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 1111 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) |
10 | 9 | pm5.32da 568 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴) ↔ (𝑦 ∈ 𝐶 ∧ 𝑦 = 𝐵))) |
11 | 7, 10 | bitr4d 271 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑦 = 𝐵 ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴))) |
12 | 11 | eubidv 2638 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (∃!𝑦 𝑦 = 𝐵 ↔ ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴))) |
13 | 4, 12 | mpbid 222 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) |
14 | df-reu 3068 | . 2 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = 𝐴 ↔ ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) | |
15 | 13, 14 | sylibr 224 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ∃!weu 2618 ∃!wreu 3063 Vcvv 3351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-reu 3068 df-v 3353 |
This theorem is referenced by: reuhyp 5024 riotaocN 35014 |
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