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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 7422. (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 3502 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ V) |
3 | eueq 3717 | . . . 4 ⊢ (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵) | |
4 | 2, 3 | sylib 218 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 𝑦 = 𝐵) |
5 | eleq1 2827 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
6 | 1, 5 | syl5ibrcom 247 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑦 = 𝐵 → 𝑦 ∈ 𝐶)) |
7 | 6 | pm4.71rd 562 | . . . . 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 1117 | . . . . . 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 282 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑦 = 𝐵 ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴))) |
12 | 11 | eubidv 2584 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (∃!𝑦 𝑦 = 𝐵 ↔ ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴))) |
13 | 4, 12 | mpbid 232 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) |
14 | df-reu 3379 | . 2 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = 𝐴 ↔ ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) | |
15 | 13, 14 | sylibr 234 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∃!weu 2566 ∃!wreu 3376 Vcvv 3478 |
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-3an 1088 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-v 3480 |
This theorem is referenced by: reuhyp 5426 riotaocN 39191 |
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