Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > vtocl2gf | Structured version Visualization version GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.) |
Ref | Expression |
---|---|
vtocl2gf.1 | ⊢ Ⅎ𝑥𝐴 |
vtocl2gf.2 | ⊢ Ⅎ𝑦𝐴 |
vtocl2gf.3 | ⊢ Ⅎ𝑦𝐵 |
vtocl2gf.4 | ⊢ Ⅎ𝑥𝜓 |
vtocl2gf.5 | ⊢ Ⅎ𝑦𝜒 |
vtocl2gf.6 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtocl2gf.7 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
vtocl2gf.8 | ⊢ 𝜑 |
Ref | Expression |
---|---|
vtocl2gf | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3212 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | vtocl2gf.3 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
3 | vtocl2gf.2 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
4 | 3 | nfel1 2779 | . . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V |
5 | vtocl2gf.5 | . . . 4 ⊢ Ⅎ𝑦𝜒 | |
6 | 4, 5 | nfim 1825 | . . 3 ⊢ Ⅎ𝑦(𝐴 ∈ V → 𝜒) |
7 | vtocl2gf.7 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
8 | 7 | imbi2d 330 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒))) |
9 | vtocl2gf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
10 | vtocl2gf.4 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
11 | vtocl2gf.6 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
12 | vtocl2gf.8 | . . . 4 ⊢ 𝜑 | |
13 | 9, 10, 11, 12 | vtoclgf 3264 | . . 3 ⊢ (𝐴 ∈ V → 𝜓) |
14 | 2, 6, 8, 13 | vtoclgf 3264 | . 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ V → 𝜒)) |
15 | 1, 14 | mpan9 486 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 Vcvv 3200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 |
This theorem is referenced by: vtocl3gf 3269 vtocl2g 3270 vtocl2gaf 3273 offval22 7253 vtocl2d 29314 fmuldfeqlem1 39814 |
Copyright terms: Public domain | W3C validator |