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Theorem erthi 7836
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 7798 . . 3 (𝜑𝐴𝑋)
42, 3erth 7834 . 2 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
51, 4mpbid 222 1 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523   class class class wbr 4685   Er wer 7784  [cec 7785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-er 7787  df-ec 7789
This theorem is referenced by:  erdisj  7837  qsel  7869  addsrmo  9932  mulsrmo  9933  qusgrp2  17580  frgpinv  18223  qustgpopn  21970  blpnfctr  22288  pi1inv  22898  pi1xfrf  22899  pi1xfr  22901  pi1xfrcnvlem  22902  pi1cof  22905  vitalilem3  23424  sconnpi1  31347
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