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Theorem erthi 8549
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 (𝜑𝑅 Er 𝑋)
erthi.2 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
erthi (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Proof of Theorem erthi
StepHypRef Expression
1 erthi.2 . 2 (𝜑𝐴𝑅𝐵)
2 erthi.1 . . 3 (𝜑𝑅 Er 𝑋)
32, 1ercl 8509 . . 3 (𝜑𝐴𝑋)
42, 3erth 8547 . 2 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
51, 4mpbid 231 1 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539   class class class wbr 5074   Er wer 8495  [cec 8496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-er 8498  df-ec 8500
This theorem is referenced by:  erdisj  8550  qsel  8585  addsrmo  10829  mulsrmo  10830  qusgrp2  18693  frgpinv  19370  qustgpopn  23271  blpnfctr  23589  pi1inv  24215  pi1xfrf  24216  pi1xfr  24218  pi1xfrcnvlem  24219  pi1cof  24222  vitalilem3  24774  sconnpi1  33201  qsalrel  40215
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