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Mirrors > Home > MPE Home > Th. List > issetf | Structured version Visualization version GIF version |
Description: A version of isset 3424 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 3424 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴) | |
2 | issetf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfeq2 2985 | . . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
4 | nfv 2015 | . . 3 ⊢ Ⅎ𝑦 𝑥 = 𝐴 | |
5 | eqeq1 2829 | . . 3 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴)) | |
6 | 3, 4, 5 | cbvexv1 2370 | . 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴) |
7 | 1, 6 | bitri 267 | 1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1658 ∃wex 1880 ∈ wcel 2166 Ⅎwnfc 2956 Vcvv 3414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-v 3416 |
This theorem is referenced by: vtoclgf 3480 spcimgft 3501 |
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