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Theorem eqvrelsym 37096
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 37088 . . . 4 ( EqvRel 𝑅 → Rel 𝑅)
4 relbrcnvg 6062 . . . 4 (Rel 𝑅 → (𝐵𝑅𝐴𝐴𝑅𝐵))
52, 3, 43syl 18 . . 3 (𝜑 → (𝐵𝑅𝐴𝐴𝑅𝐵))
61, 5mpbird 257 . 2 (𝜑𝐵𝑅𝐴)
7 eqvrelsymrel 37090 . . . 4 ( EqvRel 𝑅 → SymRel 𝑅)
8 dfsymrel2 37040 . . . . 5 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
98simplbi 499 . . . 4 ( SymRel 𝑅𝑅𝑅)
102, 7, 93syl 18 . . 3 (𝜑𝑅𝑅)
1110ssbrd 5153 . 2 (𝜑 → (𝐵𝑅𝐴𝐵𝑅𝐴))
126, 11mpd 15 1 (𝜑𝐵𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wss 3915   class class class wbr 5110  ccnv 5637  Rel wrel 5643   SymRel wsymrel 36675   EqvRel weqvrel 36680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-refrel 37003  df-symrel 37035  df-trrel 37065  df-eqvrel 37076
This theorem is referenced by:  eqvrelsymb  37097  eqvreltr4d  37100  eqvrelth  37102
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