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| 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.) | 
| Ref | Expression | 
|---|---|
| erthi.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) | 
| erthi.2 | ⊢ (𝜑 → 𝐴𝑅𝐵) | 
| Ref | Expression | 
|---|---|
| erthi | ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | erthi.2 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
| 2 | erthi.1 | . . 3 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
| 3 | 2, 1 | ercl 8757 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | 
| 4 | 2, 3 | erth 8797 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅)) | 
| 5 | 1, 4 | mpbid 232 | 1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 class class class wbr 5142 Er wer 8743 [cec 8744 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-er 8746 df-ec 8748 | 
| This theorem is referenced by: erdisj 8800 qsel 8837 addsrmo 11114 mulsrmo 11115 qusgrp2 19077 frgpinv 19783 qustgpopn 24129 blpnfctr 24447 pi1inv 25086 pi1xfrf 25087 pi1xfr 25089 pi1xfrcnvlem 25090 pi1cof 25093 vitalilem3 25646 rloccring 33275 fracfld 33311 qsdrngilem 33523 zringfrac 33583 sconnpi1 35245 qsalrel 42281 | 
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