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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 2809 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | vtocleg.1 | . . 3 ⊢ (𝑥 = 𝐴 → 𝜑) | |
3 | 2 | exlimiv 1925 | . 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 1533 ∃wex 1773 ∈ wcel 2098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-clel 2804 |
This theorem is referenced by: vtoclg 3537 vtocle 3538 spsbc 3785 snexg 5423 prex 5425 avril1 30220 bj-snexg 36421 rdgssun 36765 finxpreclem6 36783 ralssiun 36794 frege58c 43230 tz6.12i-afv2 46505 |
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