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| Mirrors > Home > MPE Home > Th. List > vtocle | Structured version Visualization version GIF version | ||
| Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.) Avoid df-clab 2715. (Revised by Wolf Lammen, 31-May-2025.) |
| Ref | Expression |
|---|---|
| vtocle.1 | ⊢ 𝐴 ∈ V |
| vtocle.2 | ⊢ (𝑥 = 𝐴 → 𝜑) |
| Ref | Expression |
|---|---|
| vtocle | ⊢ 𝜑 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vtocle.2 | . 2 ⊢ (𝑥 = 𝐴 → 𝜑) | |
| 2 | vtocle.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 3 | 2 | isseti 3498 | . 2 ⊢ ∃𝑥 𝑥 = 𝐴 |
| 4 | 1, 3 | exlimiiv 1931 | 1 ⊢ 𝜑 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-clel 2816 |
| This theorem is referenced by: vtocl 3558 zfrepclf 5291 tz6.12i 6934 eloprabga 7542 cfflb 10299 axcc3 10478 nn0ind-raph 12718 finxpreclem6 37397 ormkglobd 46890 tworepnotupword 46901 |
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