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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 2822 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
| 2 | vtocleg.1 | . . 3 ⊢ (𝑥 = 𝐴 → 𝜑) | |
| 3 | 2 | exlimiv 1929 | . 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 1539 ∃wex 1778 ∈ wcel 2107 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-clel 2815 | 
| This theorem is referenced by: vtoclg 3553 vtocleOLD 3555 spsbc 3800 snexg 5434 prex 5436 avril1 30483 bj-snexg 37036 rdgssun 37380 finxpreclem6 37398 ralssiun 37409 frege58c 43939 tz6.12i-afv2 47260 | 
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