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Theorem erthi 6592
Description: Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
erthi.1  |-  ( ph  ->  R  Er  X )
erthi.2  |-  ( ph  ->  A R B )
Assertion
Ref Expression
erthi  |-  ( ph  ->  [ A ] R  =  [ B ] R
)

Proof of Theorem erthi
StepHypRef Expression
1 erthi.2 . 2  |-  ( ph  ->  A R B )
2 erthi.1 . . 3  |-  ( ph  ->  R  Er  X )
32, 1ercl 6557 . . 3  |-  ( ph  ->  A  e.  X )
42, 3erth 6590 . 2  |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R
) )
51, 4mpbid 203 1  |-  ( ph  ->  [ A ] R  =  [ B ] R
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619   class class class wbr 3920    Er wer 6543   [cec 6544
This theorem is referenced by:  erdisj  6593  qsel  6624  th3qlem1  6650  divsgrp2  14448  frgpinv  14908  divstgpopn  17634  blpnfctr  17814  pi1inv  18382  pi1xfrf  18383  pi1xfr  18385  pi1xfrcnvlem  18386  pi1cof  18389  vitalilem3  18797  sconpi1  22941  pdiveql  25334
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-er 6546  df-ec 6548
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