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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 2838 | . 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)) |
3 | 2 | impcom 125 | 1 ⊢ ((∀𝑥 ∈ 𝐵 𝜑 ∧ 𝐴 ∈ 𝐵) → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∀wral 2455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2740 |
This theorem is referenced by: disjne 3477 seex 4336 fconstfvm 5735 fvixp 6703 ordiso2 7034 eqord1 8440 eqord2 8441 seq3caopr2 10482 bccl 10747 2clim 11309 isummulc2 11434 telfsumo2 11475 fsumparts 11478 isumshft 11498 mertenslem2 11544 mertensabs 11545 dvdsprime 12122 mgmlrid 12798 grprinvlem 12804 grpinvex 12887 issubg2m 13049 issubg4m 13053 nmzbi 13069 cnima 13723 dceqnconst 14810 dcapnconst 14811 |
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