![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 2852 | . 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 1364 ∈ wcel 2160 ∀wral 2468 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 |
This theorem is referenced by: disjne 3491 seex 4353 fconstfvm 5755 fvixp 6729 ordiso2 7064 eqord1 8470 eqord2 8471 seq3caopr2 10513 bccl 10779 2clim 11341 isummulc2 11466 telfsumo2 11507 fsumparts 11510 isumshft 11530 mertenslem2 11576 mertensabs 11577 dvdsprime 12154 mgmlrid 12855 grpinvalem 12861 grpinvex 12955 issubg2m 13128 issubg4m 13132 nmzbi 13148 cnima 14177 dceqnconst 15267 dcapnconst 15268 |
Copyright terms: Public domain | W3C validator |