Theorem erthi 8760

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 8715	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
4	2, 3	erth 8758	. 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 5114   Er wer 8700  [cec 8701

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

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 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-er 8703  df-ec 8705

This theorem is used by:  erdisj  8761  qsel  8803  addsrmo  11076  mulsrmo  11077  qusgrp2  19155  frgpinv  19865  qustgpopn  24314  blpnfctr  24630  pi1inv  25248  pi1xfrf  25249  pi1xfr  25251  pi1xfrcnvlem  25252  pi1cof  25255  vitalilem3  25806  rloccring  33622  rlocinvunit  33626  rlocisunit  33627  fracfld  33660  qsdrngilem  33807  zringfrac  33875  sconnpi1  35752  qsalrel  43050