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

Theorem ersymb 7408
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 471 . . 3  |-  ( (
ph  /\  A R B )  ->  R  Er  X )
3 simpr 467 . . 3  |-  ( (
ph  /\  A R B )  ->  A R B )
42, 3ersym 7406 . 2  |-  ( (
ph  /\  A R B )  ->  B R A )
51adantr 471 . . 3  |-  ( (
ph  /\  B R A )  ->  R  Er  X )
6 simpr 467 . . 3  |-  ( (
ph  /\  B R A )  ->  B R A )
75, 6ersym 7406 . 2  |-  ( (
ph  /\  B R A )  ->  A R B )
84, 7impbida 848 1  |-  ( ph  ->  ( A R B  <-> 
B R A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   class class class wbr 4418    Er wer 7391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4419  df-opab 4478  df-xp 4862  df-rel 4863  df-cnv 4864  df-er 7394
This theorem is referenced by:  ercnv  7415  erth  7439  erth2  7440  iiner  7466  ensymb  7648
  Copyright terms: Public domain W3C validator