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Mirrors > Home > ILE Home > Th. List > iota2df | GIF version |
Description: A condition that allows us to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.) |
Ref | Expression |
---|---|
iota2df.1 | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
iota2df.2 | ⊢ (𝜑 → ∃!𝑥𝜓) |
iota2df.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) |
iota2df.4 | ⊢ Ⅎ𝑥𝜑 |
iota2df.5 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
iota2df.6 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
iota2df | ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iota2df.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
2 | iota2df.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) | |
3 | simpr 109 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵) | |
4 | 3 | eqeq2d 2182 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((℩𝑥𝜓) = 𝑥 ↔ (℩𝑥𝜓) = 𝐵)) |
5 | 2, 4 | bibi12d 234 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((𝜓 ↔ (℩𝑥𝜓) = 𝑥) ↔ (𝜒 ↔ (℩𝑥𝜓) = 𝐵))) |
6 | iota2df.2 | . . 3 ⊢ (𝜑 → ∃!𝑥𝜓) | |
7 | iota1 5172 | . . 3 ⊢ (∃!𝑥𝜓 → (𝜓 ↔ (℩𝑥𝜓) = 𝑥)) | |
8 | 6, 7 | syl 14 | . 2 ⊢ (𝜑 → (𝜓 ↔ (℩𝑥𝜓) = 𝑥)) |
9 | iota2df.4 | . 2 ⊢ Ⅎ𝑥𝜑 | |
10 | iota2df.6 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
11 | iota2df.5 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
12 | nfiota1 5160 | . . . . 5 ⊢ Ⅎ𝑥(℩𝑥𝜓) | |
13 | 12 | a1i 9 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓)) |
14 | 13, 10 | nfeqd 2327 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓) = 𝐵) |
15 | 11, 14 | nfbid 1581 | . 2 ⊢ (𝜑 → Ⅎ𝑥(𝜒 ↔ (℩𝑥𝜓) = 𝐵)) |
16 | 1, 5, 8, 9, 10, 15 | vtocldf 2781 | 1 ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 Ⅎwnf 1453 ∃!weu 2019 ∈ wcel 2141 Ⅎwnfc 2299 ℩cio 5156 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-sn 3587 df-pr 3588 df-uni 3795 df-iota 5158 |
This theorem is referenced by: iota2d 5183 iota2 5186 riota2df 5826 |
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