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

Theorem nfcri 2252
Description: Consequence of the not-free predicate. (Note that unlike nfcr 2250, 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 2251 . 2 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
32nfi 1423 1 𝑥 𝑦𝐴
Colors of variables: wff set class
Syntax hints:  wnf 1421  wcel 1465  wnfc 2245
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-cleq 2110  df-clel 2113  df-nfc 2247
This theorem is referenced by:  clelsb3f  2262  nfnfc  2265  nfeq  2266  nfel  2267  cleqf  2282  sbabel  2284  r2alf  2429  r2exf  2430  nfrabxy  2588  cbvralf  2625  cbvrexf  2626  cbvrab  2658  rmo3f  2854  nfccdeq  2880  sbcabel  2962  cbvcsb  2979  cbvralcsf  3032  cbvrexcsf  3033  cbvreucsf  3034  cbvrabcsf  3035  dfss2f  3058  nfdif  3167  nfun  3202  nfin  3252  nfop  3691  nfiunxy  3809  nfiinxy  3810  nfiunya  3811  nfiinya  3812  cbviun  3820  cbviin  3821  iunxsngf  3860  cbvdisj  3886  nfdisjv  3888  disjiun  3894  nfmpt  3990  cbvmptf  3992  nffrfor  4240  onintrab2im  4404  tfis  4467  nfxp  4536  opeliunxp  4564  iunxpf  4657  elrnmpt1  4760  fvmptssdm  5473  nfmpo  5808  cbvmpox  5817  fmpox  6066  nffrec  6261  nfsum1  11093  nfsum  11094  fsum2dlemstep  11171  fisumcom2  11175  ctiunctlemudc  11877  ctiunctlemfo  11879
  Copyright terms: Public domain W3C validator