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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 3709 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 14-Nov-2004.) (Revised by Gino Giotto, 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 2714 | . . . 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 2636 | . . . . 5 ⊢ ∃*𝑦(𝜑 ∧ 𝑥 = 𝐴) |
4 | ancom 463 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝜑)) | |
5 | 4 | mobii 2630 | . . . . 5 ⊢ (∃*𝑦(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑)) |
6 | 3, 5 | mpbi 232 | . . . 4 ⊢ ∃*𝑦(𝑥 = 𝐴 ∧ 𝜑) |
7 | 1, 6 | mpg 1797 | . . 3 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
8 | 2euswapv 2714 | . . . 4 ⊢ (∀𝑦∃*𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑))) | |
9 | moeq 3694 | . . . . . 6 ⊢ ∃*𝑥 𝑥 = 𝐴 | |
10 | 9 | moani 2636 | . . . . 5 ⊢ ∃*𝑥(𝜑 ∧ 𝑥 = 𝐴) |
11 | 4 | mobii 2630 | . . . . 5 ⊢ (∃*𝑥(𝜑 ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
12 | 10, 11 | mpbi 232 | . . . 4 ⊢ ∃*𝑥(𝑥 = 𝐴 ∧ 𝜑) |
13 | 8, 12 | mpg 1797 | . . 3 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑)) |
14 | 7, 13 | impbii 211 | . 2 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
15 | euxfr2w.1 | . . . 4 ⊢ 𝐴 ∈ V | |
16 | biidd 264 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜑)) | |
17 | 15, 16 | ceqsexv 3538 | . . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜑) |
18 | 17 | eubii 2669 | . 2 ⊢ (∃!𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑) |
19 | 14, 18 | bitri 277 | 1 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ∃*wmo 2619 ∃!weu 2652 Vcvv 3491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-cleq 2813 df-clel 2892 |
This theorem is referenced by: euxfrw 3708 euop2 5395 |
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