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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 2727 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴) | |
2 | issetf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfeq2 2318 | . . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
4 | nfv 1515 | . . 3 ⊢ Ⅎ𝑦 𝑥 = 𝐴 | |
5 | eqeq1 2171 | . . 3 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴)) | |
6 | 3, 4, 5 | cbvex 1743 | . 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 1342 ∃wex 1479 ∈ wcel 2135 Ⅎwnfc 2293 Vcvv 2721 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 |
This theorem is referenced by: vtoclgf 2779 spcimgft 2797 spcimegft 2799 bj-vtoclgft 13497 |
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