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Mirrors > Home > MPE Home > Th. List > nfralw | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for restricted quantification. Version of nfral 3226 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 1-Sep-1999.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
nfralw.1 | ⊢ Ⅎ𝑥𝐴 |
nfralw.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfralw | ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1805 | . . 3 ⊢ Ⅎ𝑦⊤ | |
2 | nfralw.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
4 | nfralw.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
6 | 1, 3, 5 | nfraldw 3223 | . 2 ⊢ (⊤ → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑) |
7 | 6 | mptru 1544 | 1 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1538 Ⅎwnf 1784 Ⅎwnfc 2961 ∀wral 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 |
This theorem is referenced by: nfra2w 3227 rspc2 3631 sbcralt 3855 reu8nf 3860 rspc2vd 3932 raaan 4460 raaan2 4464 reusngf 4612 rexreusng 4617 reuprg0 4638 nfint 4886 nfiin 4950 disjxun 5064 nfpo 5479 nfso 5480 nffr 5529 nfse 5530 ralxpf 5717 reuop 6144 wfisg 6183 dff13f 7014 nfiso 7075 mpoeq123 7226 nfofr 7415 fiun 7644 f1iun 7645 fmpox 7765 ovmptss 7788 nfwrecs 7949 xpf1o 8679 ac6sfi 8762 nfoi 8978 scottexs 9316 scott0s 9317 lble 11593 nnwof 12315 fzrevral 12993 reuccatpfxs1 14109 rlimcld2 14935 fsum2dlem 15125 fsumcom2 15129 fprod2dlem 15334 fprodcom2 15338 gsummoncoe1 20472 cnmpt21 22279 cfilucfil 23169 ulmss 24985 fsumdvdscom 25762 dchrisumlema 26064 dchrisumlem2 26066 cnlnadjlem5 29848 rspc2daf 30231 disjabrex 30332 disjabrexf 30333 aciunf1lem 30407 fnpreimac 30416 fsumiunle 30545 ordtconnlem1 31167 esumiun 31353 bnj1366 32101 bnj1385 32104 bnj981 32222 bnj1228 32283 bnj1398 32306 bnj1445 32316 bnj1449 32320 bnj1463 32327 untsucf 32936 setinds 33023 tfisg 33055 frpoinsg 33081 frinsg 33087 nffrecs 33120 nosupbnd1 33214 poimirlem26 34933 poimirlem27 34934 indexdom 35024 filbcmb 35030 sdclem1 35033 scottexf 35461 scott0f 35462 cdleme31sn1 37532 cdlemk36 38064 setindtrs 39671 nfscott 40624 scottabf 40625 evth2f 41321 evthf 41333 uzwo4 41364 disjinfi 41503 choicefi 41512 rnmptbd2lem 41569 rnmptbdlem 41576 ssfiunibd 41625 infxrunb2 41685 supxrunb3 41721 supxrleubrnmpt 41728 allbutfiinf 41743 suprleubrnmpt 41745 infxrgelbrnmpt 41779 climinff 41941 limsupre3uzlem 42065 xlimmnfv 42164 xlimpnfv 42168 cncfshift 42206 cncficcgt0 42220 stoweidlem31 42365 stoweidlem34 42368 stoweidlem35 42369 stoweidlem51 42385 stoweidlem53 42387 stoweidlem54 42388 stoweidlem57 42391 stoweidlem59 42393 stoweidlem60 42394 fourierdlem31 42472 fourierdlem73 42513 iundjiun 42791 meaiuninc3v 42815 issmfle 43071 issmfgt 43082 issmfge 43095 smfpimcc 43131 smfsup 43137 smfinf 43141 2reu3 43358 2reu8i 43361 ichreuopeq 43684 reupr 43733 reuopreuprim 43737 |
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