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Mirrors > Home > ILE Home > Th. List > nfcri | GIF version |
Description: Consequence of the not-free predicate. (Note that unlike nfcr 2311, this does not require 𝑦 and 𝐴 to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfcri.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfcri | ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcri.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfcrii 2312 | . 2 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
3 | 2 | nfi 1462 | 1 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnf 1460 ∈ wcel 2148 Ⅎwnfc 2306 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-cleq 2170 df-clel 2173 df-nfc 2308 |
This theorem is referenced by: clelsb1f 2323 nfnfc 2326 nfeq 2327 nfel 2328 cleqf 2344 sbabel 2346 r2alf 2494 r2exf 2495 nfrabxy 2658 cbvralfw 2695 cbvrexfw 2696 cbvralf 2697 cbvrexf 2698 cbvrab 2737 rmo3f 2936 nfccdeq 2962 sbcabel 3046 cbvcsbw 3063 cbvcsb 3064 cbvralcsf 3121 cbvrexcsf 3122 cbvreucsf 3123 cbvrabcsf 3124 dfss2f 3148 nfdif 3258 nfun 3293 nfin 3343 nfop 3796 nfiunxy 3914 nfiinxy 3915 nfiunya 3916 nfiinya 3917 cbviun 3925 cbviin 3926 iunxsngf 3966 cbvdisj 3992 nfdisjv 3994 disjiun 4000 nfmpt 4097 cbvmptf 4099 nffrfor 4350 onintrab2im 4519 tfis 4584 nfxp 4655 opeliunxp 4683 iunxpf 4777 elrnmpt1 4880 fvmptssdm 5602 nfmpo 5946 cbvmpox 5955 fmpox 6203 nffrec 6399 cc3 7269 nfsum1 11366 nfsum 11367 fsum2dlemstep 11444 fisumcom2 11448 nfcprod1 11564 nfcprod 11565 cbvprod 11568 fprod2dlemstep 11632 fprodcom2fi 11636 ctiunctlemudc 12440 ctiunctlemfo 12442 |
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