![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2819 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | vtocleg.1 | . . 3 ⊢ (𝑥 = 𝐴 → 𝜑) | |
3 | 2 | exlimiv 1933 | . 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 1541 ∃wex 1781 ∈ wcel 2106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-clel 2814 |
This theorem is referenced by: vtoclg 3523 vtocle 3542 spsbc 3750 snexg 5385 prex 5387 avril1 29293 bj-snexg 35472 rdgssun 35816 finxpreclem6 35834 ralssiun 35845 frege58c 42135 tz6.12i-afv2 45407 |
Copyright terms: Public domain | W3C validator |