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

Theorem ersymb 7343
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 465 . . 3  |-  ( (
ph  /\  A R B )  ->  R  Er  X )
3 simpr 461 . . 3  |-  ( (
ph  /\  A R B )  ->  A R B )
42, 3ersym 7341 . 2  |-  ( (
ph  /\  A R B )  ->  B R A )
51adantr 465 . . 3  |-  ( (
ph  /\  B R A )  ->  R  Er  X )
6 simpr 461 . . 3  |-  ( (
ph  /\  B R A )  ->  B R A )
75, 6ersym 7341 . 2  |-  ( (
ph  /\  B R A )  ->  A R B )
84, 7impbida 832 1  |-  ( ph  ->  ( A R B  <-> 
B R A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   class class class wbr 4456    Er wer 7326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-er 7329
This theorem is referenced by:  ercnv  7350  erth  7374  erth2  7375  iiner  7401  ensymb  7582
  Copyright terms: Public domain W3C validator