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

Theorem ersymb 6921
Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
ersymb.1  |-  ( ph  ->  R  Er  X )
Assertion
Ref Expression
ersymb  |-  ( ph  ->  ( A R B  <-> 
B R A ) )

Proof of Theorem ersymb
StepHypRef Expression
1 ersymb.1 . . . 4  |-  ( ph  ->  R  Er  X )
21adantr 453 . . 3  |-  ( (
ph  /\  A R B )  ->  R  Er  X )
3 simpr 449 . . 3  |-  ( (
ph  /\  A R B )  ->  A R B )
42, 3ersym 6919 . 2  |-  ( (
ph  /\  A R B )  ->  B R A )
51adantr 453 . . 3  |-  ( (
ph  /\  B R A )  ->  R  Er  X )
6 simpr 449 . . 3  |-  ( (
ph  /\  B R A )  ->  B R A )
75, 6ersym 6919 . 2  |-  ( (
ph  /\  B R A )  ->  A R B )
84, 7impbida 807 1  |-  ( ph  ->  ( A R B  <-> 
B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   class class class wbr 4214    Er wer 6904
This theorem is referenced by:  ercnv  6928  erth  6951  erth2  6952  iiner  6978  ensymb  7157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-er 6907
  Copyright terms: Public domain W3C validator