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| Mirrors > Home > MPE Home > Th. List > euxfr2w | Structured version Visualization version GIF version | ||
| Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr2 3727 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by NM, 14-Nov-2004.) Avoid ax-13 2376. (Revised by GG, 10-Jan-2024.) | 
| Ref | Expression | 
|---|---|
| euxfr2w.1 | ⊢ 𝐴 ∈ V | 
| euxfr2w.2 | ⊢ ∃*𝑦 𝑥 = 𝐴 | 
| Ref | Expression | 
|---|---|
| euxfr2w | ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 2euswapv 2629 | . . . 4 ⊢ (∀𝑥∃*𝑦(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) | |
| 2 | euxfr2w.2 | . . . . . 6 ⊢ ∃*𝑦 𝑥 = 𝐴 | |
| 3 | 2 | moani 2552 | . . . . 5 ⊢ ∃*𝑦(𝜑 ∧ 𝑥 = 𝐴) | 
| 4 | ancom 460 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝜑)) | |
| 5 | 4 | mobii 2547 | . . . . 5 ⊢ (∃*𝑦(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑)) | 
| 6 | 3, 5 | mpbi 230 | . . . 4 ⊢ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑) | 
| 7 | 1, 6 | mpg 1796 | . . 3 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | 
| 8 | 2euswapv 2629 | . . . 4 ⊢ (∀𝑦∃*𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑))) | |
| 9 | moeq 3712 | . . . . . 6 ⊢ ∃*𝑥 𝑥 = 𝐴 | |
| 10 | 9 | moani 2552 | . . . . 5 ⊢ ∃*𝑥(𝜑 ∧ 𝑥 = 𝐴) | 
| 11 | 4 | mobii 2547 | . . . . 5 ⊢ (∃*𝑥(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑)) | 
| 12 | 10, 11 | mpbi 230 | . . . 4 ⊢ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑) | 
| 13 | 8, 12 | mpg 1796 | . . 3 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑)) | 
| 14 | 7, 13 | impbii 209 | . 2 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | 
| 15 | euxfr2w.1 | . . . 4 ⊢ 𝐴 ∈ V | |
| 16 | biidd 262 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜑)) | |
| 17 | 15, 16 | ceqsexv 3531 | . . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜑) | 
| 18 | 17 | eubii 2584 | . 2 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑) | 
| 19 | 14, 18 | bitri 275 | 1 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ∃*wmo 2537 ∃!weu 2567 Vcvv 3479 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-mo 2539 df-eu 2568 df-cleq 2728 df-clel 2815 | 
| This theorem is referenced by: euxfrw 3726 euop2 5516 | 
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