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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 3731 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by NM, 14-Nov-2004.) Avoid ax-13 2375. (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 2628 | . . . 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 2551 | . . . . 5 ⊢ ∃*𝑦(𝜑 ∧ 𝑥 = 𝐴) |
4 | ancom 460 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝜑)) | |
5 | 4 | mobii 2546 | . . . . 5 ⊢ (∃*𝑦(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑)) |
6 | 3, 5 | mpbi 230 | . . . 4 ⊢ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑) |
7 | 1, 6 | mpg 1794 | . . 3 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
8 | 2euswapv 2628 | . . . 4 ⊢ (∀𝑦∃*𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑))) | |
9 | moeq 3716 | . . . . . 6 ⊢ ∃*𝑥 𝑥 = 𝐴 | |
10 | 9 | moani 2551 | . . . . 5 ⊢ ∃*𝑥(𝜑 ∧ 𝑥 = 𝐴) |
11 | 4 | mobii 2546 | . . . . 5 ⊢ (∃*𝑥(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
12 | 10, 11 | mpbi 230 | . . . 4 ⊢ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑) |
13 | 8, 12 | mpg 1794 | . . 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 3530 | . . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜑) |
18 | 17 | eubii 2583 | . 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 1537 ∃wex 1776 ∈ wcel 2106 ∃*wmo 2536 ∃!weu 2566 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-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-mo 2538 df-eu 2567 df-cleq 2727 df-clel 2814 |
This theorem is referenced by: euxfrw 3730 euop2 5522 |
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