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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 1466 | . 2 ⊢ Ⅎ𝑥𝜑 | |
5 | nfvd 1467 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
6 | nfcvd 2229 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | iota2df 5004 | 1 ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1289 ∈ wcel 1438 ∃!weu 1948 ℩cio 4978 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-rex 2365 df-v 2621 df-sbc 2841 df-un 3003 df-sn 3452 df-pr 3453 df-uni 3654 df-iota 4980 |
This theorem is referenced by: (None) |
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