Theorem ersymb 6120

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 261	. . 3 |- ( ( ph /\ A R B ) -> R Er X )
3		simpr 103	. . 3 |- ( ( ph /\ A R B ) -> A R B )
4	2, 3	ersym 6118	. 2 |- ( ( ph /\ A R B ) -> B R A )
5	1	adantr 261	. . 3 |- ( ( ph /\ B R A ) -> R Er X )
6		simpr 103	. . 3 |- ( ( ph /\ B R A ) -> B R A )
7	5, 6	ersym 6118	. 2 |- ( ( ph /\ B R A ) -> A R B )
8	4, 7	impbida 528	1 |- ( ph -> ( A R B <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   class class class wbr 3764   Er wer 6103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-er 6106

This theorem is referenced by:  ercnv  6127  erth  6150  erth2  6151  iinerm  6178  ensymb  6260