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Theorem eqvrelsym 38607
Description: An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)
Hypotheses
Ref Expression
eqvrelsym.1 (𝜑 → EqvRel 𝑅)
eqvrelsym.2 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
eqvrelsym (𝜑𝐵𝑅𝐴)

Proof of Theorem eqvrelsym
StepHypRef Expression
1 eqvrelsym.2 . . 3 (𝜑𝐴𝑅𝐵)
2 eqvrelsym.1 . . . 4 (𝜑 → EqvRel 𝑅)
3 eqvrelrel 38599 . . . 4 ( EqvRel 𝑅 → Rel 𝑅)
4 relbrcnvg 6122 . . . 4 (Rel 𝑅 → (𝐵𝑅𝐴𝐴𝑅𝐵))
52, 3, 43syl 18 . . 3 (𝜑 → (𝐵𝑅𝐴𝐴𝑅𝐵))
61, 5mpbird 257 . 2 (𝜑𝐵𝑅𝐴)
7 eqvrelsymrel 38601 . . . 4 ( EqvRel 𝑅 → SymRel 𝑅)
8 dfsymrel2 38551 . . . . 5 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
98simplbi 497 . . . 4 ( SymRel 𝑅𝑅𝑅)
102, 7, 93syl 18 . . 3 (𝜑𝑅𝑅)
1110ssbrd 5185 . 2 (𝜑 → (𝐵𝑅𝐴𝐵𝑅𝐴))
126, 11mpd 15 1 (𝜑𝐵𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wss 3950   class class class wbr 5142  ccnv 5683  Rel wrel 5689   SymRel wsymrel 38195   EqvRel weqvrel 38200
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-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-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  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-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-refrel 38514  df-symrel 38546  df-trrel 38576  df-eqvrel 38587
This theorem is referenced by:  eqvrelsymb  38608  eqvreltr4d  38611  eqvrelth  38613
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