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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 2005 | . . . . . 6 ⊢ (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴) | |
3 | 1, 2 | ax-mp 7 | . . . . 5 ⊢ ∃𝑦 𝑥 = 𝐴 |
4 | 3 | biantrur 299 | . . . 4 ⊢ (𝜑 ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑)) |
5 | 19.41v 1856 | . . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑)) | |
6 | euxfrdc.3 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 6 | pm5.32i 447 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜓)) |
8 | 7 | exbii 1567 | . . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓)) |
9 | 4, 5, 8 | 3bitr2i 207 | . . 3 ⊢ (𝜑 ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓)) |
10 | 9 | eubii 1984 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓)) |
11 | euxfrdc.1 | . . 3 ⊢ 𝐴 ∈ V | |
12 | 1 | eumoi 2008 | . . 3 ⊢ ∃*𝑦 𝑥 = 𝐴 |
13 | 11, 12 | euxfr2dc 2838 | . 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 802 = wceq 1314 ∃wex 1451 ∈ wcel 1463 ∃!weu 1975 Vcvv 2657 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-v 2659 |
This theorem is referenced by: (None) |
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