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Theorem erthi 6183
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 6148 . . 3 (𝜑𝐴𝑋)
42, 3erth 6181 . 2 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
51, 4mpbid 139 1 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259   class class class wbr 3792   Er wer 6134  [cec 6135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-er 6137  df-ec 6139
This theorem is referenced by:  qsel  6214  th3qlem1  6239  mulcanenqec  6542  mulcanenq0ec  6601  addnq0mo  6603  mulnq0mo  6604  addsrmo  6886  mulsrmo  6887
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