Theorem ersymb 6511

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 6509	. 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 6509	. 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 3981   Er wer 6494

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-rex 2449  df-v 2727  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-br 3982  df-opab 4043  df-xp 4609  df-rel 4610  df-cnv 4611  df-er 6497

This theorem is referenced by:  ercnv  6518  erth  6541  erth2  6542  iinerm  6569  ensymb  6742