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Mirrors > Home > ILE Home > Th. List > spcimdv | GIF version |
Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
spcimdv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
spcimdv.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
spcimdv | ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcimdv.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) | |
2 | 1 | ex 108 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 → 𝜒))) |
3 | 2 | alrimiv 1754 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒))) |
4 | spcimdv.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
5 | nfv 1421 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
6 | nfcv 2178 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
7 | 5, 6 | spcimgft 2629 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜓 → 𝜒))) |
8 | 3, 4, 7 | sylc 56 | 1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 = wceq 1243 ∈ wcel 1393 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 |
This theorem is referenced by: spcdv 2638 rspcimdv 2657 |
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