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Mirrors > Home > ILE Home > Th. List > iota2d | 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)) |
Ref | Expression |
---|---|
iota2d | ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iota2df.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
2 | iota2df.2 | . 2 ⊢ (𝜑 → ∃!𝑥𝜓) | |
3 | iota2df.3 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) | |
4 | nfv 1539 | . 2 ⊢ Ⅎ𝑥𝜑 | |
5 | nfvd 1540 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
6 | nfcvd 2333 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | iota2df 5221 | 1 ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∃!weu 2038 ∈ wcel 2160 ℩cio 5194 |
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-v 2754 df-sbc 2978 df-un 3148 df-sn 3613 df-pr 3614 df-uni 3825 df-iota 5196 |
This theorem is referenced by: (None) |
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