Theorem erthi 6698

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 6661	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
4	2, 3	erth 6696	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
5	1, 4	mpbid 147	1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Syntax hints:   → wi 4   = wceq 1375   class class class wbr 4062   Er wer 6647  [cec 6648

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-sbc 3009  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-er 6650  df-ec 6652

This theorem is referenced by:  qsel  6729  th3qlem1  6754  mulcanenqec  7541  mulcanenq0ec  7600  addnq0mo  7602  mulnq0mo  7603  addsrmo  7898  mulsrmo  7899  qusgrp2  13616  blpnfctr  15078