Theorem ersymb 6549

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 276	. . 3 |- ( ( ph /\ A R B ) -> R Er X )
3		simpr 110	. . 3 |- ( ( ph /\ A R B ) -> A R B )
4	2, 3	ersym 6547	. 2 |- ( ( ph /\ A R B ) -> B R A )
5	1	adantr 276	. . 3 |- ( ( ph /\ B R A ) -> R Er X )
6		simpr 110	. . 3 |- ( ( ph /\ B R A ) -> B R A )
7	5, 6	ersym 6547	. 2 |- ( ( ph /\ B R A ) -> A R B )
8	4, 7	impbida 596	1 |- ( ph -> ( A R B <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   class class class wbr 4004   Er wer 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635  df-er 6535

This theorem is referenced by:  ercnv  6556  erth  6579  erth2  6580  iinerm  6607  ensymb  6780