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Mirrors > Home > MPE Home > Th. List > vtocleg | Structured version Visualization version GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
vtocleg.1 | ⊢ (𝑥 = 𝐴 → 𝜑) |
Ref | Expression |
---|---|
vtocleg | ⊢ (𝐴 ∈ 𝑉 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 3188 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | vtocleg.1 | . . 3 ⊢ (𝑥 = 𝐴 → 𝜑) | |
3 | 2 | exlimiv 1845 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑) |
4 | 1, 3 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = 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 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-tru 1478 df-ex 1696 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 |
This theorem is referenced by: vtocle 3255 spsbc 3415 prex 4836 avril1 26711 finxpreclem6 32409 frege58c 37235 |
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