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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvreud | Structured version Visualization version GIF version |
Description: Deduction used to change bound variables in a restricted existential uniqueness quantifier. (Contributed by ML, 27-Mar-2021.) |
Ref | Expression |
---|---|
cbvreud.1 | ⊢ Ⅎ𝑥𝜑 |
cbvreud.2 | ⊢ Ⅎ𝑦𝜑 |
cbvreud.3 | ⊢ (𝜑 → Ⅎ𝑦𝜓) |
cbvreud.4 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
cbvreud.5 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
cbvreud | ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvreud.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | cbvreud.2 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfvd 1923 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦 𝑥 ∈ 𝐴) | |
4 | cbvreud.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
5 | 3, 4 | nfand 1905 | . . 3 ⊢ (𝜑 → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜓)) |
6 | nfvd 1923 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) | |
7 | cbvreud.4 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
8 | 6, 7 | nfand 1905 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜒)) |
9 | eleq1 2825 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
10 | 9 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) |
11 | cbvreud.5 | . . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
12 | 11 | imp 410 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
13 | 10, 12 | anbi12d 634 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑦 ∈ 𝐴 ∧ 𝜒))) |
14 | 13 | ex 416 | . . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑦 ∈ 𝐴 ∧ 𝜒)))) |
15 | 1, 2, 5, 8, 14 | cbveud 35280 | . 2 ⊢ (𝜑 → (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜒))) |
16 | df-reu 3068 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
17 | df-reu 3068 | . 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜒 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜒)) | |
18 | 15, 16, 17 | 3bitr4g 317 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 ∃!weu 2567 ∃!wreu 3063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 df-mo 2539 df-eu 2568 df-cleq 2729 df-clel 2816 df-reu 3068 |
This theorem is referenced by: fvineqsneu 35319 |
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