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Mirrors > Home > ILE Home > Th. List > issetf | GIF version |
Description: A version of isset that does not require x and A to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Ref | Expression |
---|---|
issetf.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
issetf | ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isset 2687 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴) | |
2 | issetf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfeq2 2291 | . . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
4 | nfv 1508 | . . 3 ⊢ Ⅎ𝑦 𝑥 = 𝐴 | |
5 | eqeq1 2144 | . . 3 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴)) | |
6 | 3, 4, 5 | cbvex 1729 | . 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴) |
7 | 1, 6 | bitri 183 | 1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Ⅎwnfc 2266 Vcvv 2681 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 |
This theorem is referenced by: vtoclgf 2739 spcimgft 2757 spcimegft 2759 bj-vtoclgft 12971 |
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