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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elissetv | Structured version Visualization version GIF version |
Description: Version of bj-elisset 32056 with a dv condition on 𝑥, 𝑉. This proof uses only df-ex 1696, ax-gen 1713, ax-4 1728 and df-clel 2606 on top of propositional calculus. Prefer its use over bj-elisset 32056 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 | df-clel 2606 | . 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉)) | |
2 | exsimpl 1783 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | sylbi 206 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-clel 2606 |
This theorem is referenced by: bj-elisset 32056 bj-issetiv 32057 bj-ceqsaltv 32070 bj-ceqsalgv 32074 bj-vtoclg1fv 32104 bj-ru 32126 |
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