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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 3501 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
| 2 | vtocl2gf.3 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
| 3 | vtocl2gf.2 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
| 4 | 3 | nfel1 2922 | . . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V | 
| 5 | vtocl2gf.5 | . . . 4 ⊢ Ⅎ𝑦𝜒 | |
| 6 | 4, 5 | nfim 1896 | . . 3 ⊢ Ⅎ𝑦(𝐴 ∈ V → 𝜒) | 
| 7 | vtocl2gf.7 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
| 8 | 7 | imbi2d 340 | . . 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 3569 | . . 3 ⊢ (𝐴 ∈ V → 𝜓) | 
| 14 | 2, 6, 8, 13 | vtoclgf 3569 | . 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 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 Vcvv 3480 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 | 
| This theorem is referenced by: vtocl3gf 3573 vtocl2gafOLD 3580 offval22 8113 fmuldfeqlem1 45597 | 
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