Theorem erthi 8805

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 8760	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
4	2, 3	erth 8803	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
5	1, 4	mpbid 235	1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   class class class wbr 5106   Er wer 8745  [cec 8746

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-er 8748  df-ec 8750

This theorem is used by:  erdisj  8806  qsel  8848  addsrmo  11188  mulsrmo  11189  qusmgm  18896  qusmnd  19007  qusgrp2  19300  frgpinv  20011  qustgpopn  24472  blpnfctr  24788  pi1inv  25406  pi1xfrf  25407  pi1xfr  25409  pi1xfrcnvlem  25410  pi1cof  25413  vitalilem3  25964  rloccring  33908  rlocinvunit  33912  rlocisunit  33913  fracfld  33954  qsdrngilem  34102  zringfrac  34169  sconnpi1  36184  qsalrel  43496