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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-rspg | GIF version |
Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2653 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.) |
Ref | Expression |
---|---|
bj-rspg.nfa | ⊢ Ⅎ𝑥𝐴 |
bj-rspg.nfb | ⊢ Ⅎ𝑥𝐵 |
bj-rspg.nf2 | ⊢ Ⅎ𝑥𝜓 |
bj-rspg.is | ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
bj-rspg | ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-rspg.nfa | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | bj-rspg.nfb | . . 3 ⊢ Ⅎ𝑥𝐵 | |
3 | bj-rspg.nf2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | 1, 2, 3 | bj-rspgt 9925 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓))) |
5 | bj-rspg.is | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) | |
6 | 4, 5 | mpg 1340 | 1 ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 Ⅎwnf 1349 ∈ wcel 1393 Ⅎwnfc 2165 ∀wral 2306 |
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-ral 2311 df-v 2559 |
This theorem is referenced by: bj-bdfindisg 10073 bj-findisg 10105 |
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