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Theorem syl5eleqr 2522
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl5eleqr.1  |-  A  e.  B
syl5eleqr.2  |-  ( ph  ->  C  =  B )
Assertion
Ref Expression
syl5eleqr  |-  ( ph  ->  A  e.  C )

Proof of Theorem syl5eleqr
StepHypRef Expression
1 syl5eleqr.1 . 2  |-  A  e.  B
2 syl5eleqr.2 . . 3  |-  ( ph  ->  C  =  B )
32eqcomd 2440 . 2  |-  ( ph  ->  B  =  C )
41, 3syl5eleq 2521 1  |-  ( ph  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725
This theorem is referenced by:  rabsnt  3873  reusv6OLD  4726  onnev  4950  opabiota  6530  canth  6531  onnseq  6598  tfrlem16  6646  oen0  6821  nnawordex  6872  inf0  7568  cantnflt  7619  cnfcom2  7651  cnfcom3lem  7652  cnfcom3  7653  r1ordg  7696  r1val1  7704  rankr1id  7780  acacni  8012  dfacacn  8013  dfac13  8014  cda1dif  8048  ttukeylem5  8385  ttukeylem6  8386  gchac  8540  gch2  8546  gch3  8547  gchina  8566  swrds1  11779  sadcp1  12959  xpsfrnel2  13782  idfucl  14070  gsumz  14773  gsumval2  14775  frmdmnd  14796  frmd0  14797  efginvrel2  15351  efgcpbl2  15381  pgpfaclem1  15631  lbsexg  16228  dfac14  17642  acufl  17941  cnextfvval  18088  cnextcn  18090  minveclem3b  19321  minveclem4a  19323  ovollb2  19377  ovolunlem1a  19384  ovolunlem1  19385  ovoliunlem1  19390  ovoliun2  19394  ioombl1lem4  19447  uniioombllem1  19465  uniioombllem2  19467  uniioombllem6  19472  itg2monolem1  19634  itg2mono  19637  itg2cnlem1  19645  xrlimcnp  20799  efrlim  20800  cusgrares  21473  usgrcyclnl1  21619  ex-br  21731  vcoprne  22050  rge0scvg  24327  topjoin  26385  aomclem1  27120  dfac21  27132  frlmlbs  27217  pclfinN  30634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-cleq 2428  df-clel 2431
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