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Mirrors > Home > MPE Home > Th. List > issetf | Structured version Visualization version GIF version |
Description: A version of isset 3207 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 3207 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴) | |
2 | issetf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfeq2 2780 | . . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
4 | nfv 1843 | . . 3 ⊢ Ⅎ𝑦 𝑥 = 𝐴 | |
5 | eqeq1 2626 | . . 3 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴)) | |
6 | 3, 4, 5 | cbvex 2272 | . 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴) |
7 | 1, 6 | bitri 264 | 1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Ⅎwnfc 2751 Vcvv 3200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 |
This theorem is referenced by: vtoclgf 3264 spcimgft 3284 |
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