![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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.) |
Ref | Expression |
---|---|
vtocle.1 | ⊢ 𝐴 ∈ V |
vtocle.2 | ⊢ (𝑥 = 𝐴 → 𝜑) |
Ref | Expression |
---|---|
vtocle | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtocle.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | vtocle.2 | . . 3 ⊢ (𝑥 = 𝐴 → 𝜑) | |
3 | 2 | vtocleg 3529 | . 2 ⊢ (𝐴 ∈ V → 𝜑) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-cleq 2791 df-clel 2870 |
This theorem is referenced by: zfrepclf 5162 tz6.12i 6671 eloprabga 7240 cfflb 9670 axcc3 9849 nn0ind-raph 12070 finxpreclem6 34813 |
Copyright terms: Public domain | W3C validator |