Theorem erthi 6635

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 6598	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
4	2, 3	erth 6633	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
5	1, 4	mpbid 147	1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Syntax hints:   → wi 4   = wceq 1364   class class class wbr 4029   Er wer 6584  [cec 6585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-er 6587  df-ec 6589

This theorem is referenced by:  qsel  6666  th3qlem1  6691  mulcanenqec  7446  mulcanenq0ec  7505  addnq0mo  7507  mulnq0mo  7508  addsrmo  7803  mulsrmo  7804  qusgrp2  13183  blpnfctr  14607