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Mirrors > Home > ILE Home > Th. List > rspccva | GIF version |
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
rspcv.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rspccva | ⊢ ((∀𝑥 ∈ 𝐵 𝜑 ∧ 𝐴 ∈ 𝐵) → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcv.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | 1 | rspcv 2789 | . 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)) |
3 | 2 | impcom 124 | 1 ⊢ ((∀𝑥 ∈ 𝐵 𝜑 ∧ 𝐴 ∈ 𝐵) → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 ∀wral 2417 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 |
This theorem is referenced by: disjne 3421 seex 4265 fconstfvm 5646 grprinvlem 5973 fvixp 6605 ordiso2 6928 eqord1 8269 eqord2 8270 seq3caopr2 10286 bccl 10545 2clim 11102 isummulc2 11227 telfsumo2 11268 fsumparts 11271 isumshft 11291 mertenslem2 11337 mertensabs 11338 dvdsprime 11839 cnima 12428 dceqnconst 13423 |
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