![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 2805 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑧))) | |
2 | rmo4.1 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | equequ1 1645 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) | |
4 | equcom 1639 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦) | |
5 | 3, 4 | syl6bb 194 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑧 = 𝑦)) |
6 | 2, 5 | imbi12d 232 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜓 → 𝑧 = 𝑦))) |
7 | 6 | cbvralv 2590 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑧) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦)) |
8 | 7 | rexbii 2385 | . . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑧) ↔ ∃𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦)) |
9 | equequ1 1645 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 ↔ 𝑥 = 𝑦)) | |
10 | 9 | imbi2d 228 | . . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝜓 → 𝑧 = 𝑦) ↔ (𝜓 → 𝑥 = 𝑦))) |
11 | 10 | ralbidv 2380 | . . . . 5 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
12 | 11 | cbvrexv 2591 | . . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) |
13 | 8, 12 | bitri 182 | . . 3 ⊢ (∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑧) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) |
14 | 13 | anbi2i 445 | . 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 182 | 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 102 ↔ wb 103 ∀wral 2359 ∃wrex 2360 ∃!wreu 2361 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |