Theorem nfcri 2369

Description: Consequence of the not-free predicate. (Note that unlike nfcr 2367, this does not require 𝑦 and 𝐴 to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfcri.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcri	⊢ Ⅎ𝑥 𝑦 ∈ 𝐴

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem nfcri

Step	Hyp	Ref	Expression
1		nfcri.1	. . 3 ⊢ Ⅎ𝑥𝐴
2	1	nfcrii 2368	. 2 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
3	2	nfi 1511	1 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴

Syntax hints:  Ⅎwnf 1509   ∈ wcel 2202  Ⅎwnfc 2362

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2364

This theorem is referenced by:  clelsb1f  2379  nfnfc  2382  nfeq  2383  nfel  2384  cleqf  2400  sbabel  2402  r2alf  2550  r2exf  2551  nfrabw  2715  cbvralfw  2757  cbvrexfw  2758  cbvralf  2759  cbvrexf  2760  cbvrab  2801  rmo3f  3004  nfccdeq  3030  sbcabel  3115  cbvcsbw  3132  cbvcsb  3133  cbvralcsf  3191  cbvrexcsf  3192  cbvreucsf  3193  cbvrabcsf  3194  dfss2f  3219  nfdif  3330  nfun  3365  nfin  3415  nfop  3883  nfiunxy  4001  nfiinxy  4002  nfiunya  4003  nfiinya  4004  cbviun  4012  cbviin  4013  iunxsngf  4053  cbvdisj  4079  nfdisjv  4081  disjiun  4088  nfmpt  4186  cbvmptf  4188  nffrfor  4451  onintrab2im  4622  tfis  4687  nfxp  4758  opeliunxp  4787  iunxpf  4884  elrnmpt1  4989  fvmptssdm  5740  nfmpo  6100  cbvmpox  6109  fmpox  6374  nffrec  6605  cc3  7530  nfsum1  11977  nfsum  11978  fsum2dlemstep  12056  fisumcom2  12060  nfcprod1  12176  nfcprod  12177  cbvprod  12180  fprod2dlemstep  12244  fprodcom2fi  12248  ctiunctlemudc  13119  ctiunctlemfo  13121