Theorem ersymb 6451

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

Step	Hyp	Ref	Expression
1		ersymb.1	. . . 4 |- ( ph -> R Er X )
2	1	adantr 274	. . 3 |- ( ( ph /\ A R B ) -> R Er X )
3		simpr 109	. . 3 |- ( ( ph /\ A R B ) -> A R B )
4	2, 3	ersym 6449	. 2 |- ( ( ph /\ A R B ) -> B R A )
5	1	adantr 274	. . 3 |- ( ( ph /\ B R A ) -> R Er X )
6		simpr 109	. . 3 |- ( ( ph /\ B R A ) -> B R A )
7	5, 6	ersym 6449	. 2 |- ( ( ph /\ B R A ) -> A R B )
8	4, 7	impbida 586	1 |- ( ph -> ( A R B <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   class class class wbr 3937   Er wer 6434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-er 6437

This theorem is referenced by:  ercnv  6458  erth  6481  erth2  6482  iinerm  6509  ensymb  6682