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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elissetv | Structured version Visualization version GIF version |
Description: Version of bj-elisset 34194 with a disjoint variable condition on 𝑥, 𝑉. This proof uses only df-ex 1781, ax-gen 1796, ax-4 1810 and df-clel 2895 on top of propositional calculus. Prefer its use over bj-elisset 34194 when sufficient. (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-elissetv | ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfclel 2896 | . 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉)) | |
2 | exsimpl 1869 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | sylbi 219 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-clel 2895 |
This theorem is referenced by: bj-elisset 34194 bj-issetiv 34195 bj-ceqsaltv 34205 bj-ceqsalgv 34209 bj-spcimdvv 34214 bj-vtoclg1fv 34237 bj-vtoclg 34238 bj-ru 34257 |
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