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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 229 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) |
4 | 1, 3 | rspcimdv 3612 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 |
This theorem is referenced by: rspcdv2 3617 rspcv 3618 ralxfrd 5414 ralxfrd2 5418 reuop 6315 suppofss1d 8228 suppofss2d 8229 zindd 12717 wrd2ind 14758 ismri2dad 17682 mreexd 17687 mreexexlemd 17689 catcocl 17730 catass 17731 moni 17784 subccocl 17896 funcco 17922 fullfo 17966 fthf1 17971 nati 18010 mndind 18854 ringurd 20203 idsrngd 20874 mpomulcn 24905 fsumdvdsmul 27253 uspgr2wlkeq 29679 crctcshwlkn0lem4 29843 crctcshwlkn0lem5 29844 wwlknllvtx 29876 0enwwlksnge1 29894 wlkiswwlks2lem5 29903 clwlkclwwlklem2a 30027 clwlkclwwlklem2 30029 clwwisshclwws 30044 clwwlkinwwlk 30069 umgr2cwwk2dif 30093 wrdt2ind 32923 mgccole1 32965 mgccole2 32966 mgcmnt1 32967 mgcmntco 32969 dfmgc2lem 32970 chnind 32985 1arithufdlem3 33554 dfufd2 33558 fedgmullem2 33658 constrconj 33750 zart0 33840 zarcmplem 33842 esumcvg 34067 inelcarsg 34293 carsgclctunlem1 34299 orvcelel 34451 signsply0 34545 onint1 36432 qsalrel 42260 ismnushort 44297 ralbinrald 47072 fargshiftfva 47368 reupr 47447 evengpop3 47723 evengpoap3 47724 snlindsntorlem 48316 |
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