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Mirrors > Home > MPE Home > Th. List > rspcdv | Structured version Visualization version GIF version |
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
rspcdv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
rspcdv.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rspcdv | ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcdv.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | rspcdv.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 2 | biimpd 228 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) |
4 | 1, 3 | rspcimdv 3602 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 |
This theorem is referenced by: rspcdv2 3607 rspcv 3608 ralxfrd 5406 ralxfrd2 5410 reuop 6292 suppofss1d 8191 suppofss2d 8192 zindd 12667 wrd2ind 14677 ismri2dad 17585 mreexd 17590 mreexexlemd 17592 catcocl 17633 catass 17634 moni 17687 subccocl 17799 funcco 17825 fullfo 17867 fthf1 17872 nati 17910 mndind 18745 ringurd 20079 idsrngd 20613 mpomulcn 24605 uspgr2wlkeq 29158 crctcshwlkn0lem4 29322 crctcshwlkn0lem5 29323 wwlknllvtx 29355 0enwwlksnge1 29373 wlkiswwlks2lem5 29382 clwlkclwwlklem2a 29506 clwlkclwwlklem2 29508 clwwisshclwws 29523 clwwlkinwwlk 29548 umgr2cwwk2dif 29572 wrdt2ind 32372 mgccole1 32415 mgccole2 32416 mgcmnt1 32417 mgcmntco 32419 dfmgc2lem 32420 fedgmullem2 32991 zart0 33145 zarcmplem 33147 esumcvg 33370 inelcarsg 33596 carsgclctunlem1 33602 orvcelel 33754 signsply0 33848 onint1 35637 qsalrel 41368 ismnushort 43362 ralbinrald 46129 fargshiftfva 46410 reupr 46489 evengpop3 46765 evengpoap3 46766 isomgrsym 46803 snlindsntorlem 47239 |
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