Theorem erthi 8728

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	⊢ (𝜑 → 𝑅 Er 𝑋)
erthi.2	⊢ (𝜑 → 𝐴𝑅𝐵)

Assertion

Ref	Expression
erthi	⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Proof of Theorem erthi

Step	Hyp	Ref	Expression
1		erthi.2	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
2		erthi.1	. . 3 ⊢ (𝜑 → 𝑅 Er 𝑋)
3	2, 1	ercl 8683	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
4	2, 3	erth 8726	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
5	1, 4	mpbid 234	1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Syntax hints:   → wi 4   = wceq 1559   class class class wbr 5097   Er wer 8668  [cec 8669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-er 8671  df-ec 8673

This theorem is referenced by:  erdisj  8729  qsel  8771  addsrmo  11024  mulsrmo  11025  qusgrp2  19090  frgpinv  19794  qustgpopn  24167  blpnfctr  24483  pi1inv  25101  pi1xfrf  25102  pi1xfr  25104  pi1xfrcnvlem  25105  pi1cof  25108  vitalilem3  25659  rloccring  33412  rlocinvunit  33416  rlocisunit  33417  fracfld  33455  qsdrngilem  33642  zringfrac  33710  sconnpi1  35549  qsalrel  42817