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Mirrors > Home > MPE Home > Th. List > ralsn | Structured version Visualization version GIF version |
Description: Convert a universal quantification restricted to a singleton to a substitution. (Contributed by NM, 27-Apr-2009.) |
Ref | Expression |
---|---|
ralsn.1 | ⊢ 𝐴 ∈ V |
ralsn.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ralsn | ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | ralsn.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | ralsng 4679 | . 2 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 ∀wral 3058 Vcvv 3477 {csn 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-v 3479 df-sn 4631 |
This theorem is referenced by: xpord2indlem 8170 xpord3inddlem 8177 naddcllem 8712 naddasslem1 8730 naddasslem2 8731 elixpsn 8975 frfi 9318 dffi3 9468 ssttrcl 9752 ttrclss 9757 ttrclselem2 9763 fseqenlem1 10061 fpwwe2lem12 10679 hashbc 14488 hashf1lem1 14490 eqs1 14646 cshw1 14856 rpnnen2lem11 16256 drsdirfi 18362 0subg 19181 0subgOLD 19182 0subgALT 19600 efgsp1 19769 dprd2da 20076 lbsextlem4 21180 rnglidl0 21256 ply1coe 22317 mat0dimcrng 22491 txkgen 23675 xkoinjcn 23710 isufil2 23931 ust0 24243 prdsxmetlem 24393 prdsbl 24519 finiunmbl 25592 xrlimcnp 27025 chtub 27270 2sqlem10 27486 dchrisum0flb 27568 pntpbnd1 27644 conway 27858 etasslt 27872 slerec 27878 bday1s 27890 madebdaylemlrcut 27951 precsexlem9 28253 zscut 28407 nohalf 28421 halfcut 28430 addhalfcut 28433 usgr1e 29276 nbgr2vtx1edg 29381 nbuhgr2vtx1edgb 29383 wlkl1loop 29670 crctcshwlkn0lem7 29845 2pthdlem1 29959 rusgrnumwwlkl1 29997 clwwlkccatlem 30017 clwwlkn2 30072 clwwlkel 30074 clwwlkwwlksb 30082 1wlkdlem4 30168 h1deoi 31577 bnj149 34867 subfacp1lem5 35168 cvmlift2lem1 35286 cvmlift2lem12 35298 lindsenlbs 37601 poimirlem28 37634 poimirlem32 37638 heibor1lem 37795 nadd1suc 43381 usgrexmpl1lem 47915 usgrexmpl2lem 47920 |
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