ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfcri GIF version

Theorem nfcri 2250
Description: Consequence of the not-free predicate. (Note that unlike nfcr 2248, this does not require 𝑦 and 𝐴 to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
nfcri.1 𝑥𝐴
Assertion
Ref Expression
nfcri 𝑥 𝑦𝐴
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem nfcri
StepHypRef Expression
1 nfcri.1 . . 3 𝑥𝐴
21nfcrii 2249 . 2 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
32nfi 1421 1 𝑥 𝑦𝐴
Colors of variables: wff set class
Syntax hints:  wnf 1419  wcel 1463  wnfc 2243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-cleq 2108  df-clel 2111  df-nfc 2245
This theorem is referenced by:  clelsb3f  2260  nfnfc  2263  nfeq  2264  nfel  2265  cleqf  2280  sbabel  2282  r2alf  2427  r2exf  2428  nfrabxy  2586  cbvralf  2623  cbvrexf  2624  cbvrab  2656  rmo3f  2852  nfccdeq  2878  sbcabel  2960  cbvcsb  2977  cbvralcsf  3030  cbvrexcsf  3031  cbvreucsf  3032  cbvrabcsf  3033  dfss2f  3056  nfdif  3165  nfun  3200  nfin  3250  nfop  3689  nfiunxy  3807  nfiinxy  3808  nfiunya  3809  nfiinya  3810  cbviun  3818  cbviin  3819  iunxsngf  3858  cbvdisj  3884  nfdisjv  3886  disjiun  3892  nfmpt  3988  cbvmptf  3990  nffrfor  4238  onintrab2im  4402  tfis  4465  nfxp  4534  opeliunxp  4562  iunxpf  4655  elrnmpt1  4758  fvmptssdm  5471  nfmpo  5806  cbvmpox  5815  fmpox  6064  nffrec  6259  nfsum1  11065  nfsum  11066  fsum2dlemstep  11143  fisumcom2  11147  ctiunctlemudc  11845  ctiunctlemfo  11847
  Copyright terms: Public domain W3C validator