MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ersymb Structured version   Visualization version   GIF version

Theorem ersymb 8723
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 480 . . 3 ((𝜑𝐴𝑅𝐵) → 𝑅 Er 𝑋)
3 simpr 484 . . 3 ((𝜑𝐴𝑅𝐵) → 𝐴𝑅𝐵)
42, 3ersym 8721 . 2 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
51adantr 480 . . 3 ((𝜑𝐵𝑅𝐴) → 𝑅 Er 𝑋)
6 simpr 484 . . 3 ((𝜑𝐵𝑅𝐴) → 𝐵𝑅𝐴)
75, 6ersym 8721 . 2 ((𝜑𝐵𝑅𝐴) → 𝐴𝑅𝐵)
84, 7impbida 798 1 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   class class class wbr 5148   Er wer 8706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-er 8709
This theorem is referenced by:  ercnv  8730  erth  8758  erth2  8759  iiner  8789  ensymb  9004
  Copyright terms: Public domain W3C validator