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Theorem elec 6935
Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)
Hypotheses
Ref Expression
elec.1  |-  A  e. 
_V
elec.2  |-  B  e. 
_V
Assertion
Ref Expression
elec  |-  ( A  e.  [ B ] R 
<->  B R A )

Proof of Theorem elec
StepHypRef Expression
1 elec.1 . 2  |-  A  e. 
_V
2 elec.2 . 2  |-  B  e. 
_V
3 elecg 6934 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  e.  [ B ] R  <->  B R A ) )
41, 2, 3mp2an 654 1  |-  ( A  e.  [ B ] R 
<->  B R A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1725   _Vcvv 2948   class class class wbr 4204   [cec 6894
This theorem is referenced by:  ecid  6960  sylow2alem2  15240  sylow2a  15241  sylow2blem1  15242  efgval2  15344  efgrelexlemb  15370  efgcpbllemb  15375  frgpnabllem1  15472  tgpconcomp  18130  divstgphaus  18140  vitalilem2  19489  vitalilem3  19490  isbndx  26428  prtlem10  26651  prtlem19  26664  prter3  26668
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-cnv 4877  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-ec 6898
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