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Mirrors > Home > ILE Home > Th. List > euxfrdc | GIF version |
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.) |
Ref | Expression |
---|---|
euxfrdc.1 | ⊢ 𝐴 ∈ V |
euxfrdc.2 | ⊢ ∃!𝑦 𝑥 = 𝐴 |
euxfrdc.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
euxfrdc | ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euxfrdc.2 | . . . . . 6 ⊢ ∃!𝑦 𝑥 = 𝐴 | |
2 | euex 2043 | . . . . . 6 ⊢ (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ ∃𝑦 𝑥 = 𝐴 |
4 | 3 | biantrur 301 | . . . 4 ⊢ (𝜑 ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑)) |
5 | 19.41v 1889 | . . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑)) | |
6 | euxfrdc.3 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 6 | pm5.32i 450 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜓)) |
8 | 7 | exbii 1592 | . . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓)) |
9 | 4, 5, 8 | 3bitr2i 207 | . . 3 ⊢ (𝜑 ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓)) |
10 | 9 | eubii 2022 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓)) |
11 | euxfrdc.1 | . . 3 ⊢ 𝐴 ∈ V | |
12 | 1 | eumoi 2046 | . . 3 ⊢ ∃*𝑦 𝑥 = 𝐴 |
13 | 11, 12 | euxfr2dc 2906 | . 2 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦𝜓)) |
14 | 10, 13 | syl5bb 191 | 1 ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 DECID wdc 824 = wceq 1342 ∃wex 1479 ∃!weu 2013 ∈ wcel 2135 Vcvv 2721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-v 2723 |
This theorem is referenced by: (None) |
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