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Theorem ersymb 7113
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 7111 . 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 7111 . 2  |-  ( (
ph  /\  B R A )  ->  A R B )
84, 7impbida 828 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 4290    Er wer 7096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-br 4291  df-opab 4349  df-xp 4844  df-rel 4845  df-cnv 4846  df-er 7099
This theorem is referenced by:  ercnv  7120  erth  7143  erth2  7144  iiner  7170  ensymb  7355
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