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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 2233 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
3 | vtoclgaf.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | 2, 3 | nfim 1505 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓) |
5 | eleq1 2145 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
6 | vtoclgaf.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 5, 6 | imbi12d 232 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))) |
8 | vtoclgaf.4 | . . 3 ⊢ (𝑥 ∈ 𝐵 → 𝜑) | |
9 | 1, 4, 7, 8 | vtoclgf 2668 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝜓)) |
10 | 9 | pm2.43i 48 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1285 Ⅎwnf 1390 ∈ wcel 1434 Ⅎwnfc 2210 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2614 |
This theorem is referenced by: vtoclga 2675 ssiun2s 3748 tfis 4361 fvmptf 5340 fmptco 5406 prmind2 10882 |
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