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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-rspg | GIF version |
Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2781 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 12982 | . 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 1427 | 1 ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 Ⅎwnf 1436 ∈ wcel 1480 Ⅎwnfc 2266 ∀wral 2414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-v 2683 |
This theorem is referenced by: bj-bdfindisg 13135 bj-findisg 13167 |
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