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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 2333 | . 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 5872 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ℩crio 5847 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-v 2754 df-sbc 2978 df-un 3148 df-sn 3613 df-pr 3614 df-uni 3825 df-iota 5193 df-riota 5848 |
This theorem is referenced by: f1ocnvfv3 5881 caucvgrelemrec 11015 sqrt0 11040 sqrtsq 11080 dfgcd3 12038 |
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