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Theorem ersymb 8313
 Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
ersymb.1 (𝜑𝑅 Er 𝑋)
Assertion
Ref Expression
ersymb (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))

Proof of Theorem ersymb
StepHypRef Expression
1 ersymb.1 . . . 4 (𝜑𝑅 Er 𝑋)
21adantr 484 . . 3 ((𝜑𝐴𝑅𝐵) → 𝑅 Er 𝑋)
3 simpr 488 . . 3 ((𝜑𝐴𝑅𝐵) → 𝐴𝑅𝐵)
42, 3ersym 8311 . 2 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
51adantr 484 . . 3 ((𝜑𝐵𝑅𝐴) → 𝑅 Er 𝑋)
6 simpr 488 . . 3 ((𝜑𝐵𝑅𝐴) → 𝐵𝑅𝐴)
75, 6ersym 8311 . 2 ((𝜑𝐵𝑅𝐴) → 𝐴𝑅𝐵)
84, 7impbida 800 1 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   class class class wbr 5032   Er wer 8296 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-xp 5530  df-rel 5531  df-cnv 5532  df-er 8299 This theorem is referenced by:  ercnv  8320  erth  8348  erth2  8349  iiner  8379  ensymb  8575
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