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Theorem erthi 8332
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 8292 . . 3 (𝜑𝐴𝑋)
42, 3erth 8330 . 2 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
51, 4mpbid 234 1 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530   class class class wbr 5057   Er wer 8278  [cec 8279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-er 8281  df-ec 8283
This theorem is referenced by:  erdisj  8333  qsel  8368  addsrmo  10487  mulsrmo  10488  qusgrp2  18209  frgpinv  18882  qustgpopn  22720  blpnfctr  23038  pi1inv  23648  pi1xfrf  23649  pi1xfr  23651  pi1xfrcnvlem  23652  pi1cof  23655  vitalilem3  24203  sconnpi1  32474  qsalrel  39110
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