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Mirrors > Home > ILE Home > Th. List > iota2 | GIF version |
Description: The unique element such that 𝜑. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
iota2.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
iota2 | ⊢ ((𝐴 ∈ 𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2763 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | simpl 109 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝐴 ∈ V) | |
3 | simpr 110 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → ∃!𝑥𝜑) | |
4 | iota2.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | adantl 277 | . . 3 ⊢ (((𝐴 ∈ V ∧ ∃!𝑥𝜑) ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓)) |
6 | nfv 1539 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V | |
7 | nfeu1 2049 | . . . 4 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
8 | 6, 7 | nfan 1576 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ ∃!𝑥𝜑) |
9 | nfvd 1540 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝜓) | |
10 | nfcvd 2333 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝐴) | |
11 | 2, 3, 5, 8, 9, 10 | iota2df 5221 | . 2 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴)) |
12 | 1, 11 | sylan 283 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∃!weu 2038 ∈ wcel 2160 Vcvv 2752 ℩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: iotam 5227 pczpre 12329 pcdiv 12334 |
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