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Mirrors > Home > ILE Home > Th. List > riota5 | GIF version |
Description: A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.) |
Ref | Expression |
---|---|
riota5.1 | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
riota5.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵)) |
Ref | Expression |
---|---|
riota5 | ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcvd 2282 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
2 | riota5.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
3 | riota5.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵)) | |
4 | 1, 2, 3 | riota5f 5754 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ℩crio 5729 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-v 2688 df-sbc 2910 df-un 3075 df-sn 3533 df-pr 3534 df-uni 3737 df-iota 5088 df-riota 5730 |
This theorem is referenced by: f1ocnvfv3 5763 caucvgrelemrec 10751 sqrt0 10776 sqrtsq 10816 dfgcd3 11698 |
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