ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ersymb GIF version

Theorem ersymb 6120
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 261 . . 3 ((𝜑𝐴𝑅𝐵) → 𝑅 Er 𝑋)
3 simpr 103 . . 3 ((𝜑𝐴𝑅𝐵) → 𝐴𝑅𝐵)
42, 3ersym 6118 . 2 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
51adantr 261 . . 3 ((𝜑𝐵𝑅𝐴) → 𝑅 Er 𝑋)
6 simpr 103 . . 3 ((𝜑𝐵𝑅𝐴) → 𝐵𝑅𝐴)
75, 6ersym 6118 . 2 ((𝜑𝐵𝑅𝐴) → 𝐴𝑅𝐵)
84, 7impbida 528 1 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   class class class wbr 3764   Er wer 6103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-er 6106
This theorem is referenced by:  ercnv  6127  erth  6150  erth2  6151  iinerm  6178  ensymb  6260
  Copyright terms: Public domain W3C validator