Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > vtocld | Structured version Visualization version GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
vtocld.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
vtocld.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
vtocld.3 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
vtocld | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtocld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | vtocld.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | vtocld.3 | . 2 ⊢ (𝜑 → 𝜓) | |
4 | nfv 1906 | . 2 ⊢ Ⅎ𝑥𝜑 | |
5 | nfcvd 2975 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | nfvd 1907 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
7 | 1, 2, 3, 4, 5, 6 | vtocldf 3553 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-ex 1772 df-nf 1776 df-cleq 2811 df-clel 2890 df-nfc 2960 |
This theorem is referenced by: vtocl2d 3555 lmatfval 30978 lmatcl 30980 dvgrat 40521 dfatbrafv2b 43321 fnbrafv2b 43324 |
Copyright terms: Public domain | W3C validator |