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Mirrors > Home > ILE Home > Th. List > riota2f | GIF version |
Description: This theorem shows a condition that allows us to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
riota2f.1 | ⊢ Ⅎ𝑥𝐵 |
riota2f.2 | ⊢ Ⅎ𝑥𝜓 |
riota2f.3 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
riota2f | ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riota2f.1 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
2 | 1 | nfel1 2343 | . 2 ⊢ Ⅎ𝑥 𝐵 ∈ 𝐴 |
3 | 1 | a1i 9 | . 2 ⊢ (𝐵 ∈ 𝐴 → Ⅎ𝑥𝐵) |
4 | riota2f.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
5 | 4 | a1i 9 | . 2 ⊢ (𝐵 ∈ 𝐴 → Ⅎ𝑥𝜓) |
6 | id 19 | . 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴) | |
7 | riota2f.3 | . . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
8 | 7 | adantl 277 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 = 𝐵) → (𝜑 ↔ 𝜓)) |
9 | 2, 3, 5, 6, 8 | riota2df 5868 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 Ⅎwnf 1471 ∈ wcel 2160 Ⅎwnfc 2319 ∃!wreu 2470 ℩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-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: riota2 5870 riotaprop 5871 riotass2 5874 riotass 5875 |
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