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Theorem syl6eleqr 2128
Description: A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
Hypotheses
Ref Expression
syl6eleqr.1 (φA B)
syl6eleqr.2 𝐶 = B
Assertion
Ref Expression
syl6eleqr (φA 𝐶)

Proof of Theorem syl6eleqr
StepHypRef Expression
1 syl6eleqr.1 . 2 (φA B)
2 syl6eleqr.2 . . 3 𝐶 = B
32eqcomi 2041 . 2 B = 𝐶
41, 3syl6eleq 2127 1 (φA 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033
This theorem is referenced by:  brelrng  4508  elabrex  5340  fliftel1  5377  ovidig  5560  unielxp  5742  2oconcl  5961  ecopqsi  6097  eroprf  6135  addclnq  6359  mulclnq  6360  recexnq  6374  ltexnqq  6391  prarloclemarch  6401  prarloclemarch2  6402  nnnq  6405  nqnq0  6423  addclnq0  6433  mulclnq0  6434  nqpnq0nq  6435  prarloclemlt  6475  prarloclemlo  6476  prarloclemcalc  6484  nqprm  6524  cauappcvgprlem2  6631  addclsr  6661  mulclsr  6662  pitonnlem2  6723  axaddcl  6730  axmulcl  6732  uztrn2  8246  eluz2nn  8267  peano2uzs  8283
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