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Mirrors > Home > ILE Home > Th. List > eqvisset | GIF version |
Description: A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 2743 and issetri 2746. (Contributed by BJ, 27-Apr-2019.) |
Ref | Expression |
---|---|
eqvisset | ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2740 | . 2 ⊢ 𝑥 ∈ V | |
2 | eleq1 2240 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V)) | |
3 | 1, 2 | mpbii 148 | 1 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-v 2739 |
This theorem is referenced by: elxp5 5112 xpsnen 6814 fival 6962 |
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