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Mirrors > Home > ILE Home > Th. List > reu7 | GIF version |
Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
Ref | Expression |
---|---|
rmo4.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
reu7 | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reu3 2902 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑧))) | |
2 | rmo4.1 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | equequ1 1692 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) | |
4 | equcom 1686 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦) | |
5 | 3, 4 | bitrdi 195 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑧 = 𝑦)) |
6 | 2, 5 | imbi12d 233 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜓 → 𝑧 = 𝑦))) |
7 | 6 | cbvralv 2680 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑧) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦)) |
8 | 7 | rexbii 2464 | . . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑧) ↔ ∃𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦)) |
9 | equequ1 1692 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 ↔ 𝑥 = 𝑦)) | |
10 | 9 | imbi2d 229 | . . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝜓 → 𝑧 = 𝑦) ↔ (𝜓 → 𝑥 = 𝑦))) |
11 | 10 | ralbidv 2457 | . . . . 5 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
12 | 11 | cbvrexv 2681 | . . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) |
13 | 8, 12 | bitri 183 | . . 3 ⊢ (∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑧) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) |
14 | 13 | anbi2i 453 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑧)) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
15 | 1, 14 | bitri 183 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wral 2435 ∃wrex 2436 ∃!wreu 2437 |
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-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 |
This theorem is referenced by: (None) |
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