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| Mirrors > Home > ILE Home > Th. List > vtoclgaf | GIF version | ||
| Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.) |
| Ref | Expression |
|---|---|
| vtoclgaf.1 | ⊢ Ⅎ𝑥𝐴 |
| vtoclgaf.2 | ⊢ Ⅎ𝑥𝜓 |
| vtoclgaf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
| vtoclgaf.4 | ⊢ (𝑥 ∈ 𝐵 → 𝜑) |
| Ref | Expression |
|---|---|
| vtoclgaf | ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vtoclgaf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 2 | 1 | nfel1 2359 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
| 3 | vtoclgaf.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 4 | 2, 3 | nfim 1595 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓) |
| 5 | eleq1 2268 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
| 6 | vtoclgaf.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 7 | 5, 6 | imbi12d 234 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))) |
| 8 | vtoclgaf.4 | . . 3 ⊢ (𝑥 ∈ 𝐵 → 𝜑) | |
| 9 | 1, 4, 7, 8 | vtoclgf 2831 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝜓)) |
| 10 | 9 | pm2.43i 49 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ↔ wb 105 = wceq 1373 Ⅎwnf 1483 ∈ wcel 2176 Ⅎwnfc 2335 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 |
| This theorem depends on definitions: df-bi 117 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-v 2774 |
| This theorem is referenced by: vtoclga 2839 ssiun2s 3971 tfis 4631 fvmptf 5672 fmptco 5746 prmind2 12442 |
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