![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > vtoclef | Structured version Visualization version GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
vtoclef.1 | ⊢ Ⅎ𝑥𝜑 |
vtoclef.2 | ⊢ 𝐴 ∈ V |
vtoclef.3 | ⊢ (𝑥 = 𝐴 → 𝜑) |
Ref | Expression |
---|---|
vtoclef | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtoclef.2 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | isseti 3484 | . 2 ⊢ ∃𝑥 𝑥 = 𝐴 |
3 | vtoclef.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
4 | vtoclef.3 | . . 3 ⊢ (𝑥 = 𝐴 → 𝜑) | |
5 | 3, 4 | exlimi 2202 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑) |
6 | 2, 5 | ax-mp 5 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∃wex 1773 Ⅎwnf 1777 ∈ wcel 2098 Vcvv 3468 |
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 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-nf 1778 df-clel 2804 |
This theorem is referenced by: vtoclf 3546 nn0ind-raph 12663 rdgssun 36765 finxpreclem2 36777 |
Copyright terms: Public domain | W3C validator |