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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 3492 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | vtocl2gf.3 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
3 | vtocl2gf.2 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
4 | 3 | nfel1 2918 | . . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V |
5 | vtocl2gf.5 | . . . 4 ⊢ Ⅎ𝑦𝜒 | |
6 | 4, 5 | nfim 1898 | . . 3 ⊢ Ⅎ𝑦(𝐴 ∈ V → 𝜒) |
7 | vtocl2gf.7 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
8 | 7 | imbi2d 339 | . . 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 3555 | . . 3 ⊢ (𝐴 ∈ V → 𝜓) |
14 | 2, 6, 8, 13 | vtoclgf 3555 | . 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ V → 𝜒)) |
15 | 1, 14 | mpan9 506 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 Ⅎwnf 1784 ∈ wcel 2105 Ⅎwnfc 2882 Vcvv 3473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-v 3475 |
This theorem is referenced by: vtocl3gf 3562 vtocl2gaf 3568 offval22 8077 fmuldfeqlem1 44598 |
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